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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Injective function</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">"Injective" redirects here. For other uses, see <a href="Injective_module" title="Injective module">Injective module</a> and <a href="Injective_object" title="Injective object">Injective object</a>.</div>
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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="letter-spacing:0.0125em; background-color:#FFCC99"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></th></tr><tr><td class="sidebar-image"><span class="texhtml texhtml-big" style="font-size:250%;"><i>x</i> ↦ <i>f</i> (<i>x</i>)</span></td></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
<a href="History_of_the_function_concept" title="History of the function concept">History of the function concept</a></th></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
Types by <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Codomain" title="Codomain">codomain</a></th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Boolean-valued_function" title="Boolean-valued function"><span class="texhtml"><span title="arbitrary set"><var>X</var></span> → <span title="Codomain of Booleans">𝔹</span></span></a></li>
<li><a href="Ordered_pair" title="Ordered pair">
<span class="texhtml"><span title="Domain of Booleans">𝔹</span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Boolean_function" title="Boolean function">
<span class="texhtml"><span title="several Boolean variables">𝔹<sup><var>n</var></sup></span>
→ <span title="Codomain of natural numbers"><var>X</var></span></span></a></li>
<li><a href="Integer-valued_function" title="Integer-valued function">
<span class="texhtml"><span title="arbitrary set"><var>X</var></span>
→ <span title="integers">ℤ</span></span></a></li>
<li><a href="Sequence" title="Sequence">
<span class="texhtml"><span title="integers">ℤ</span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Real-valued_function" title="Real-valued function">
<span class="texhtml"><span title="arbitrary set"><var>X</var></span>
→ <span title="real numbers">ℝ</span></span></a></li>
<li><a href="Function_of_a_real_variable" title="Function of a real variable">
<span class="texhtml"><span title="real numbers">ℝ</span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Function_of_several_real_variables" title="Function of several real variables">
<span class="texhtml"><span title="real coordinate (or Euclidean) space">ℝ<sup><var>n</var></sup></span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Complex-valued_function" class="mw-redirect" title="Complex-valued function">
<span class="texhtml"><span title="arbitrary set"><var>X</var></span>
→ <span title="complex numbers">ℂ</span></span></a></li>
<li><a href="Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable">
<span class="texhtml"><span title="complex numbers">ℂ</span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Function_of_several_complex_variables" title="Function of several complex variables">
<span class="texhtml"><span title="complex coordinate space">ℂ<sup><var>n</var></sup></span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
<a href="List_of_types_of_functions" title="List of types of functions">Classes/properties</a> </th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>
<li><a href="Identity_function" title="Identity function">Identity</a></li>
<li><a href="Linear_map" title="Linear map">Linear</a></li>
<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>
<li><a href="Rational_function" title="Rational function">Rational</a></li>
<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>
<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>
<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>
<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>
<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>
<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>
<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
Constructions</th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>
<li><a href="Function_composition" title="Function composition">Composition</a></li>
<li><a href="Lambda_calculus" title="Lambda calculus">λ</a></li>
<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
Generalizations </th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>
<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>
<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>
<li><a href="Partial_function" title="Partial function">Partial</a></li>
<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>
<li><a href="Function_space" title="Function space">Space</a></li>
<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>
<li><a href="Morphism" title="Morphism">Morphism</a></li>
<li><a href="Functor" title="Functor">Functor</a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
<a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></th></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>injective function</b> (also known as <b>injection</b>, or <b>one-to-one function</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> ) is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml"><i>f</i></span> that maps <a href="Distinct_(mathematics)" class="mw-redirect" title="Distinct (mathematics)">distinct</a> elements of its domain to distinct elements of its codomain; that is, <span class="texhtml"><i>x</i><sub>1</sub> ≠ <i>x</i><sub>2</sub></span> implies <span class="texhtml"><i>f</i>(<i>x</i><sub>1</sub>) ≠ <i>f</i>(<i>x</i><sub>2</sub>)</span> (equivalently by <a href="Contraposition" title="Contraposition">contraposition</a>, <span class="texhtml"><i>f</i>(<i>x</i><sub>1</sub>) = <i>f</i>(<i>x</i><sub>2</sub>)</span> implies <span class="texhtml"><i>x</i><sub>1</sub> = <i>x</i><sub>2</sub></span>). In other words, every element of the function's <a href="Codomain" title="Codomain">codomain</a> is the <a href="Image_(mathematics)" title="Image (mathematics)">image</a> of <em>at most</em> one element of its <a href="Domain_of_a_function" title="Domain of a function">domain</a>.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The term <em>one-to-one function</em> must not be confused with <em>one-to-one correspondence</em> that refers to <a href="Bijective_function" class="mw-redirect" title="Bijective function">bijective functions</a>, which are functions such that each element in the codomain is an image of <i>exactly one</i> element in the domain.
</p><p>A <a href="Homomorphism" title="Homomorphism">homomorphism</a> between <a href="Algebraic_structure" title="Algebraic structure">algebraic structures</a> is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for <a href="Vector_space" title="Vector space">vector spaces</a>, an <em>injective homomorphism</em> is also called a <em><a href="Monomorphism" title="Monomorphism">monomorphism</a></em>. However, in the more general context of <a href="Category_theory" title="Category theory">category theory</a>, the definition of a monomorphism differs from that of an injective homomorphism.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This is thus a theorem that they are equivalent for algebraic structures; see <a href="Homomorphism#Monomorphism" title="Homomorphism">Homomorphism § Monomorphism</a> for more details.
</p><p>A function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> that is not injective is sometimes called many-to-one.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<span class="skin-invert"></span>
<div role="note" class="hatnote navigation-not-searchable">Further information on notation: <a href="Function_(mathematics)#Notation" title="Function (mathematics)">Function (mathematics) § Notation</a></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> be a function whose domain is a set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>
</semantics>
</math></span><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"></span> The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is said to be <b>injective</b> provided that for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>
</semantics>
</math></span><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"></span> if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)=f(b),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(a)=f(b),}</annotation>
</semantics>
</math></span><img src="./74f93203ccd370bf7eeed99c1330dfc51965489d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.148ex; height:2.843ex;" alt="{\displaystyle f(a)=f(b),}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=b}</annotation>
</semantics>
</math></span><img src="./1956b03d1314c7071ac1f45ed7b1e29422dcfcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}" loading="lazy"></span>; that is, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)=f(b)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(a)=f(b)}</annotation>
</semantics>
</math></span><img src="./f70209bf143b8417feef2aed98b2e86bc8f447e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.502ex; height:2.843ex;" alt="{\displaystyle f(a)=f(b)}" loading="lazy"></span> implies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mi>b</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=b.}</annotation>
</semantics>
</math></span><img src="./f6053fe8070065a7d8818843b82d98ac9ff3708d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.973ex; height:2.176ex;" alt="{\displaystyle a=b.}" loading="lazy"></span> Equivalently, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq b,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>≠<!-- ≠ --></mo>
<mi>b</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\neq b,}</annotation>
</semantics>
</math></span><img src="./6d8f802fbf115d615e2b153d1c7bb5ede24970b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.973ex; height:2.676ex;" alt="{\displaystyle a\neq b,}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)\neq f(b)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>≠<!-- ≠ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(a)\neq f(b)}</annotation>
</semantics>
</math></span><img src="./59dee0fe7888c91cbaf98c08843aff1cbe7cb46e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.502ex; height:2.843ex;" alt="{\displaystyle f(a)\neq f(b)}" loading="lazy"></span> in the <a href="Contraposition" title="Contraposition">contrapositive</a> statement.
</p><p>Symbolically,<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mi>X</mi>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mspace width="thickmathspace"></mspace>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>a</mi>
<mo>=</mo>
<mi>b</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,}</annotation>
</semantics>
</math></span></span>
which is logically equivalent to the <a href="Contraposition" title="Contraposition">contrapositive</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mi>X</mi>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mspace width="thickmathspace"></mspace>
<mi>a</mi>
<mo>≠<!-- ≠ --></mo>
<mi>b</mi>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>≠<!-- ≠ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).}</annotation>
</semantics>
</math></span></span>An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\rightarrowtail B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">↣<!-- ↣ --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:A\rightarrowtail B}</annotation>
</semantics>
</math></span><img src="./ac4c83384513983813d9555b122bce4e40911fb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.595ex; height:2.509ex;" alt="{\displaystyle f:A\rightarrowtail B}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\hookrightarrow B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">↪<!-- ↪ --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:A\hookrightarrow B}</annotation>
</semantics>
</math></span><img src="./c22e26b503616995b93fa3c70272853861247f28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.63ex; height:2.509ex;" alt="{\displaystyle f:A\hookrightarrow B}" loading="lazy"></span>), although some authors specifically reserve ↪ for an <a href="Inclusion_map" title="Inclusion map">inclusion map</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<p><i>For visual examples, readers are directed to the <a href="#Gallery">gallery section.</a></i>
</p>
<ul><li>For any set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and any subset <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>⊆<!-- ⊆ --></mo>
<mi>X</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S\subseteq X,}</annotation>
</semantics>
</math></span><img src="./bcee64dbb80f9474630b849a15567dc2acd95608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle S\subseteq X,}" loading="lazy"></span> the <a href="Inclusion_map" title="Inclusion map">inclusion map</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S\to X}</annotation>
</semantics>
</math></span><img src="./5566629251250ba644683a256f3ae6b6ec516d8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.093ex; height:2.176ex;" alt="{\displaystyle S\to X}" loading="lazy"></span> (which sends any element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>∈<!-- ∈ --></mo>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s\in S}</annotation>
</semantics>
</math></span><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy"></span> to itself) is injective. In particular, the <a href="Identity_function" title="Identity function">identity function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation>
</semantics>
</math></span><img src="./83d8a6029587ee9b365bdeab1e2f4b7c469b0219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle X\to X}" loading="lazy"></span> is always injective (and in fact bijective).</li>
<li>If the domain of a function is the <a href="Empty_set" title="Empty set">empty set</a>, then the function is the <a href="Empty_function" class="mw-redirect" title="Empty function">empty function</a>, which is injective.</li>
<li>If the domain of a function has one element (that is, it is a <a href="Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a>), then the function is always injective.</li>
<li>The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"></span> defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=2x+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=2x+1}</annotation>
</semantics>
</math></span><img src="./10ca6b62bf1326a2e8672de9d2a8bfa95240fd76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.011ex; height:2.843ex;" alt="{\displaystyle f(x)=2x+1}" loading="lazy"></span> is injective.</li>
<li>The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} \to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} \to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./bdfd1e16b7f932cdc2716a1b6bbe345089b250cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.023ex; height:2.509ex;" alt="{\displaystyle g:\mathbb {R} \to \mathbb {R} }" loading="lazy"></span> defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=x^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(x)=x^{2}}</annotation>
</semantics>
</math></span><img src="./92746066d0381ea6189ffc725768840f81d83ba3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.737ex; height:3.176ex;" alt="{\displaystyle g(x)=x^{2}}" loading="lazy"></span> is <em>not</em> injective, because (for example) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(1)=1=g(-1).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(1)=1=g(-1).}</annotation>
</semantics>
</math></span><img src="./592342091f081a40bc94abaa83feeb42c0e56871.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.99ex; height:2.843ex;" alt="{\displaystyle g(1)=1=g(-1).}" loading="lazy"></span> However, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is redefined so that its domain is the non-negative real numbers [0,+∞), then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is injective.</li>
<li>The <a href="Exponential_function" title="Exponential function">exponential function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp :\mathbb {R} \to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>exp</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \exp :\mathbb {R} \to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./c7205d1220e96f17d06c4cb20c130902102813c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.46ex; height:2.509ex;" alt="{\displaystyle \exp :\mathbb {R} \to \mathbb {R} }" loading="lazy"></span> defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x)=e^{x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>exp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \exp(x)=e^{x}}</annotation>
</semantics>
</math></span><img src="./13cc0e0007b0b0baf6553e5cd4ea883f030cc03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.046ex; height:2.843ex;" alt="{\displaystyle \exp(x)=e^{x}}" loading="lazy"></span> is injective (but not <a href="Surjective_function" title="Surjective function">surjective</a>, as no real value maps to a negative number).</li>
<li>The <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a> function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln :(0,\infty )\to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ln</mi>
<mo>:</mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ln :(0,\infty )\to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./4f7939196d78dbee36b0285b1a7e89ff11fe8ed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.498ex; height:2.843ex;" alt="{\displaystyle \ln :(0,\infty )\to \mathbb {R} }" loading="lazy"></span> defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \ln x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\mapsto \ln x}</annotation>
</semantics>
</math></span><img src="./edc10f79eb59e9e9101dc09a42aa247b574bbae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.6ex; height:2.176ex;" alt="{\displaystyle x\mapsto \ln x}" loading="lazy"></span> is injective.</li>
<li>The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} \to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} \to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./bdfd1e16b7f932cdc2716a1b6bbe345089b250cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.023ex; height:2.509ex;" alt="{\displaystyle g:\mathbb {R} \to \mathbb {R} }" loading="lazy"></span> defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=x^{n}-x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(x)=x^{n}-x}</annotation>
</semantics>
</math></span><img src="./3b3a16eecb50ad77934008103cb5c227d7a9557e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.072ex; height:2.843ex;" alt="{\displaystyle g(x)=x^{n}-x}" loading="lazy"></span> is not injective, since, for example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(0)=g(1)=0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(0)=g(1)=0.}</annotation>
</semantics>
</math></span><img src="./5568087e15c4a13c1c5df2d2f014cdd618409c2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.182ex; height:2.843ex;" alt="{\displaystyle g(0)=g(1)=0.}" loading="lazy"></span></li></ul>
<p>More generally, when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> are both the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation>
</semantics>
</math></span><img src="./0522388d36b55de7babe4bbfc49475eaf590c2bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}" loading="lazy"></span> then an injective function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"></span> is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the <em><a href="Horizontal_line_test" title="Horizontal line test">horizontal line test</a></em>.<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Injections_can_be_undone">Injections can be undone</h2></div>
<p>Functions with <a href="Inverse_function#Left_and_right_inverses" title="Inverse function">left inverses</a> are always injections. That is, given <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y,}</annotation>
</semantics>
</math></span><img src="./43b6b077a3059ca728f62c163fec3d93b8429769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.23ex; height:2.509ex;" alt="{\displaystyle f:X\to Y,}" loading="lazy"></span> if there is a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:</mo>
<mi>Y</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g:Y\to X}</annotation>
</semantics>
</math></span><img src="./10a9844999dbfd6d1ba60a6d5d37779df277a74f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.421ex; height:2.509ex;" alt="{\displaystyle g:Y\to X}" loading="lazy"></span> such that for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>
</semantics>
</math></span><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(f(x))=x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(f(x))=x}</annotation>
</semantics>
</math></span><img src="./d12c5b0b3b9b020bfc9f5e330c074b809cc54ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle g(f(x))=x}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is injective. The proof is that
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)=f(b)\rightarrow g(f(a))=g(f(b))\rightarrow a=b.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<mi>a</mi>
<mo>=</mo>
<mi>b</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(a)=f(b)\rightarrow g(f(a))=g(f(b))\rightarrow a=b.}</annotation>
</semantics>
</math></span></span>
</p><p>In this case, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is called a <a href="Retract_(category_theory)" class="mw-redirect" title="Retract (category theory)">retraction</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f.}</annotation>
</semantics>
</math></span><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"></span> Conversely, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is called a <a href="Retract_(category_theory)" class="mw-redirect" title="Retract (category theory)">section</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g.}</annotation>
</semantics>
</math></span><img src="./23a3f421f58ef3bc6f9ec70e883e1496ff871e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g.}" loading="lazy"></span>
</p><p>Conversely, every injection <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> with a non-empty domain has a left inverse <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>. It can be defined by choosing an element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> in the domain of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and setting <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(y)}</annotation>
</semantics>
</math></span><img src="./e05f288372d2eb8e3ac42c0a76cf1f7c4093e2f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.081ex; height:2.843ex;" alt="{\displaystyle g(y)}" loading="lazy"></span> to the unique element of the pre-image <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}[y]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{-1}[y]}</annotation>
</semantics>
</math></span><img src="./c5ffa43f26ddc0c27fedee54ea4051661d56fa21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.102ex; height:3.176ex;" alt="{\displaystyle f^{-1}[y]}" loading="lazy"></span> (if it is non-empty) or to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> (otherwise).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p><p>The left inverse <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is not necessarily an <a href="Inverse_function" title="Inverse function">inverse</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f,}</annotation>
</semantics>
</math></span><img src="./9e9687ea22c0f310582e97ee5f6c6a5fca28203d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f,}" loading="lazy"></span> because the composition in the other order, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∘<!-- ∘ --></mo>
<mi>g</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\circ g,}</annotation>
</semantics>
</math></span><img src="./a5598900fe5a4f86148e55e822703066b5b5e576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.236ex; height:2.509ex;" alt="{\displaystyle f\circ g,}" loading="lazy"></span> may differ from the identity on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>
</semantics>
</math></span><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"></span> In other words, an injective function can be "reversed" by a left inverse, but is not necessarily <a href="Inverse_function" title="Inverse function">invertible</a>, which requires that the function is bijective.
</p>
<div class="mw-heading mw-heading2"><h2 id="Injections_may_be_made_invertible">Injections may be made invertible</h2></div>
<p>In fact, to turn an injective function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> into a bijective (hence invertible) function, it suffices to replace its codomain <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> by its actual image <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=f(X).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J=f(X).}</annotation>
</semantics>
</math></span><img src="./164a92b3fbc1d38dfd309d21f203d347817361bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.285ex; height:2.843ex;" alt="{\displaystyle J=f(X).}" loading="lazy"></span> That is, let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:X\to J}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g:X\to J}</annotation>
</semantics>
</math></span><img src="./781ea2bf06f3805d478d5e7172fb493374bce977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.119ex; height:2.509ex;" alt="{\displaystyle g:X\to J}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(x)=f(x)}</annotation>
</semantics>
</math></span><img src="./3b98e24d7454419b763a9c9fff1d9dec79c0eee3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle g(x)=f(x)}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>
</semantics>
</math></span><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span>; then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is bijective. Indeed, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> can be factored as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {In} _{J,Y}\circ g,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>In</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>J</mi>
<mo>,</mo>
<mi>Y</mi>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>g</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {In} _{J,Y}\circ g,}</annotation>
</semantics>
</math></span><img src="./1fac993b682afc729143be9f6d91135fc248e4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.428ex; height:2.843ex;" alt="{\displaystyle \operatorname {In} _{J,Y}\circ g,}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {In} _{J,Y}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>In</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>J</mi>
<mo>,</mo>
<mi>Y</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {In} _{J,Y}}</annotation>
</semantics>
</math></span><img src="./405bd52ee0e2e57490350338c2308d8b958c7d9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.116ex; height:2.843ex;" alt="{\displaystyle \operatorname {In} _{J,Y}}" loading="lazy"></span> is the <a href="Inclusion_function" class="mw-redirect" title="Inclusion function">inclusion function</a> from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J}</annotation>
</semantics>
</math></span><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>
</semantics>
</math></span><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"></span>
</p><p>More generally, injective <a href="Partial_function" title="Partial function">partial functions</a> are called <a href="Partial_bijection" class="mw-redirect" title="Partial bijection">partial bijections</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Other_properties">Other properties</h2></div>
<div role="note" class="hatnote navigation-not-searchable">See also: <a href="List_of_set_identities_and_relations#Functions_and_sets" title="List of set identities and relations">List of set identities and relations § Functions and sets</a></div>
<span class="skin-invert"></span>
<ul><li>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> are both injective then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∘<!-- ∘ --></mo>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation>
</semantics>
</math></span><img src="./b2f61ca7838709fbae07dce9c0d513770f10cfae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" loading="lazy"></span> is injective.</li>
<li>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>∘<!-- ∘ --></mo>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation>
</semantics>
</math></span><img src="./10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}" loading="lazy"></span> is injective, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is injective (but <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> need not be).</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> is injective if and only if, given any functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g,}</annotation>
</semantics>
</math></span><img src="./81f2986cd965e404a1ee33ec84baee5c43da47fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g,}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:W\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>:</mo>
<mi>W</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h:W\to X}</annotation>
</semantics>
</math></span><img src="./1138eea341e4d40a5499215a80b1b28d888f6550.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.305ex; height:2.176ex;" alt="{\displaystyle h:W\to X}" loading="lazy"></span> whenever <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=f\circ h,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∘<!-- ∘ --></mo>
<mi>g</mi>
<mo>=</mo>
<mi>f</mi>
<mo>∘<!-- ∘ --></mo>
<mi>h</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\circ g=f\circ h,}</annotation>
</semantics>
</math></span><img src="./1587a6dbef5803f0ebccea56cd106c11a721ddbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.147ex; height:2.509ex;" alt="{\displaystyle f\circ g=f\circ h,}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=h.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>=</mo>
<mi>h</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g=h.}</annotation>
</semantics>
</math></span><img src="./477dc92ea9e2ddfa1dddac5d889a48673c64504e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.2ex; height:2.509ex;" alt="{\displaystyle g=h.}" loading="lazy"></span> In other words, injective functions are precisely the <a href="Monomorphism" title="Monomorphism">monomorphisms</a> in the <a href="Category_theory" title="Category theory">category</a> <b><a href="Category_of_sets" title="Category of sets">Set</a></b> of sets.</li>
<li>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> is injective and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> is a <a href="Subset" title="Subset">subset</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>
</semantics>
</math></span><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(f(A))=A.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{-1}(f(A))=A.}</annotation>
</semantics>
</math></span><img src="./997b1672bcdd26297211a84e614ecbea62421282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.782ex; height:3.176ex;" alt="{\displaystyle f^{-1}(f(A))=A.}" loading="lazy"></span> Thus, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> can be recovered from its <a href="Image_(function)" class="mw-redirect" title="Image (function)">image</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(A).}</annotation>
</semantics>
</math></span><img src="./43483089a73dddf063590aee1b1fc95df748c3ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.478ex; height:2.843ex;" alt="{\displaystyle f(A).}" loading="lazy"></span></li>
<li>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> is injective and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> are both subsets of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>
</semantics>
</math></span><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A\cap B)=f(A)\cap f(B).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩<!-- ∩ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>∩<!-- ∩ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(A\cap B)=f(A)\cap f(B).}</annotation>
</semantics>
</math></span><img src="./ee62806f3bf6885374947eabbcd36570c733b375.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.188ex; height:2.843ex;" alt="{\displaystyle f(A\cap B)=f(A)\cap f(B).}" loading="lazy"></span></li>
<li>Every function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:W\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>:</mo>
<mi>W</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h:W\to Y}</annotation>
</semantics>
</math></span><img src="./c77f47d16e85fb2cf89905121e4b9894df227af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.099ex; height:2.176ex;" alt="{\displaystyle h:W\to Y}" loading="lazy"></span> can be decomposed as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=f\circ g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>=</mo>
<mi>f</mi>
<mo>∘<!-- ∘ --></mo>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h=f\circ g}</annotation>
</semantics>
</math></span><img src="./a14345f3938c66b1fde08e18d8b301a59c5775a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.027ex; height:2.509ex;" alt="{\displaystyle h=f\circ g}" loading="lazy"></span> for a suitable injection <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and surjection <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g.}</annotation>
</semantics>
</math></span><img src="./23a3f421f58ef3bc6f9ec70e883e1496ff871e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g.}" loading="lazy"></span> This decomposition is unique <a href="Up_to_isomorphism" class="mw-redirect" title="Up to isomorphism">up to isomorphism</a>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> may be thought of as the <a href="Inclusion_function" class="mw-redirect" title="Inclusion function">inclusion function</a> of the range <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(W)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>W</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h(W)}</annotation>
</semantics>
</math></span><img src="./a6e0012d5623e2ef3a9ef19432bcd45c26951c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.584ex; height:2.843ex;" alt="{\displaystyle h(W)}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> as a subset of the codomain <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h.}</annotation>
</semantics>
</math></span><img src="./10d298611ab61576b6db29d9b50b6af8f12910fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.986ex; height:2.176ex;" alt="{\displaystyle h.}" loading="lazy"></span></li>
<li>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> is an injective function, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> has at least as many elements as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>
</semantics>
</math></span><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"></span> in the sense of <a href="Cardinal_number" title="Cardinal number">cardinal numbers</a>. In particular, if, in addition, there is an injection from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>
</semantics>
</math></span><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> have the same cardinal number. (This is known as the <a href="Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem" class="mw-redirect" title="Cantor–Bernstein–Schroeder theorem">Cantor–Bernstein–Schroeder theorem</a>.)</li>
<li>If both <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> are <a href="Finite_set" title="Finite set">finite</a> with the same number of elements, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> is injective if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is surjective (in which case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is bijective).</li>
<li>An injective function which is a homomorphism between two algebraic structures is an <a href="Embedding" title="Embedding">embedding</a>.</li>
<li>Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is injective can be decided by only considering the graph (and not the codomain) of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f.}</annotation>
</semantics>
</math></span><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"></span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Proving_that_functions_are_injective">Proving that functions are injective</h2></div>
<p>A proof that a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is injective depends on how the function is presented and what properties the function holds.
For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=f(y),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=f(y),}</annotation>
</semantics>
</math></span><img src="./a1e86c77e956a4a1452ae973b15dfccfe8564506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.406ex; height:2.843ex;" alt="{\displaystyle f(x)=f(y),}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mi>y</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=y.}</annotation>
</semantics>
</math></span><img src="./45a717cd4f8417e789930877c1dfcd62b1300ce1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.009ex;" alt="{\displaystyle x=y.}" loading="lazy"></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>
</p><p>Here is an example:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=2x+3}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=2x+3}</annotation>
</semantics>
</math></span></span>
</p><p>Proof: Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y.}</annotation>
</semantics>
</math></span><img src="./b918aeefba8721a6732102a5848bd4238615ec55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.23ex; height:2.509ex;" alt="{\displaystyle f:X\to Y.}" loading="lazy"></span> Suppose <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=f(y).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=f(y).}</annotation>
</semantics>
</math></span><img src="./a1779b3b82f8ea31cb8753ba439d40d53a41df60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.406ex; height:2.843ex;" alt="{\displaystyle f(x)=f(y).}" loading="lazy"></span> So <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x+3=2y+3}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo>=</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2x+3=2y+3}</annotation>
</semantics>
</math></span><img src="./db032f3a0b2e4495531588a057a2f61c4d427598.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.914ex; height:2.509ex;" alt="{\displaystyle 2x+3=2y+3}" loading="lazy"></span> implies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x=2y,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>y</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2x=2y,}</annotation>
</semantics>
</math></span><img src="./40b84f8dd0cd6498e2d37ee51b165d39ece3a212.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.555ex; height:2.509ex;" alt="{\displaystyle 2x=2y,}" loading="lazy"></span> which implies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mi>y</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=y.}</annotation>
</semantics>
</math></span><img src="./45a717cd4f8417e789930877c1dfcd62b1300ce1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.009ex;" alt="{\displaystyle x=y.}" loading="lazy"></span> Therefore, it follows from the definition that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is injective.
</p><p>There are multiple other methods of proving that a function is injective. For example, in calculus if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is a linear transformation it is sufficient to show that the kernel of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> contains only the zero vector. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.
</p><p>A graphical approach for a real-valued function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> of a real variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> is the <a href="Horizontal_line_test" title="Horizontal line test">horizontal line test</a>. If every horizontal line intersects the curve of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> in at most one point, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is injective or one-to-one.
</p>
<div class="mw-heading mw-heading2"><h2 id="Gallery">Gallery</h2></div>
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<li class="gallerybox" style="width: 215px">
<div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">An <b>injective</b> non-surjective function (injection, not a bijection)</div>
</li>
<li class="gallerybox" style="width: 215px">
<div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">An <b>injective</b> surjective function (bijection)</div>
</li>
<li class="gallerybox" style="width: 215px">
<div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">A non-injective surjective function (surjection, not a bijection)</div>
</li>
<li class="gallerybox" style="width: 215px">
<div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">A non-injective non-surjective function (also not a bijection)</div>
</li>
</ul></div></div></div>
<div class="mod-gallery mod-gallery-default mod-gallery-center"><div class="main"><div><ul class="nochecker bordered-images whitebg gallery mw-gallery-traditional" style="max-width: 669px;">
<li class="gallerybox" style="width: 215px">
<div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">Not an injective function. Here <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>
</semantics>
</math></span><img src="./f70b2694445a5901b24338a2e7a7e58f02a72a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>
</semantics>
</math></span><img src="./2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy"></span> are subsets of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,Y_{1}}</annotation>
</semantics>
</math></span><img src="./f28ed1dcb20ba4a3216205fea49ac63717d3144f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.419ex; height:2.509ex;" alt="{\displaystyle X,Y_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{2}}</annotation>
</semantics>
</math></span><img src="./6e6a0ca6752e3b9040b24de6f0c99b6099b0c861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle Y_{2}}" loading="lazy"></span> are subsets of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span>: for two regions where the function is not injective because more than one domain <a href="Element_(mathematics)" title="Element (mathematics)">element</a> can map to a single range element. That is, it is possible for <em>more than one</em> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> to map to the <em>same</em> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>
</semantics>
</math></span><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"></span></div>
</li>
<li class="gallerybox" style="width: 215px">
<div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">Making functions injective. The previous function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> can be reduced to one or more injective functions (say) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X_{1}\to Y_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X_{1}\to Y_{1}}</annotation>
</semantics>
</math></span><img src="./619b978a679df6e6b3579419dbcbcb20f9da2038.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.213ex; height:2.509ex;" alt="{\displaystyle f:X_{1}\to Y_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X_{2}\to Y_{2},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X_{2}\to Y_{2},}</annotation>
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</math></span><img src="./d5c7df7ede8642cdd3364c15385c00bbdd84a679.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.86ex; height:2.509ex;" alt="{\displaystyle f:X_{2}\to Y_{2},}" loading="lazy"></span> shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> has not changed – only the domain and range. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>
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</math></span><img src="./f70b2694445a5901b24338a2e7a7e58f02a72a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>
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</math></span><img src="./2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy"></span> are subsets of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,Y_{1}}</annotation>
</semantics>
</math></span><img src="./f28ed1dcb20ba4a3216205fea49ac63717d3144f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.419ex; height:2.509ex;" alt="{\displaystyle X,Y_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{2}}</annotation>
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</math></span><img src="./6e6a0ca6752e3b9040b24de6f0c99b6099b0c861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle Y_{2}}" loading="lazy"></span> are subsets of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span>: for two regions where the initial function can be made injective so that one domain element can map to a single range element. That is, only one <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> maps to one <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>
</semantics>
</math></span><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"></span></div>
</li>
<li class="gallerybox" style="width: 215px">
<div class="thumb" style="width: 210px; height: 210px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">Injective functions. Diagramatic interpretation in the <a href="Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a>, defined by the <a href="Map_(mathematics)" title="Map (mathematics)">mapping</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y,}</annotation>
</semantics>
</math></span><img src="./43b6b077a3059ca728f62c163fec3d93b8429769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.23ex; height:2.509ex;" alt="{\displaystyle f:X\to Y,}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=f(x),}</annotation>
</semantics>
</math></span><img src="./c9867a6ecb3cc19e19e0af39fb46523e69e616c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle y=f(x),}" loading="lazy"></span> <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X=}</annotation>
</semantics>
</math></span><img src="./318e24a850188dfafb7530dec3a0f31c3aa8c9da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.433ex; height:2.176ex;" alt="{\displaystyle X=}" loading="lazy"></span> domain of function</span>, <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y=}</annotation>
</semantics>
</math></span><img src="./7f5ec90f224c20ef874332d97185f01168dbdcf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:4.227ex; height:2.009ex;" alt="{\displaystyle Y=}" loading="lazy"></span> <a href="Range_of_a_function" title="Range of a function">range of function</a></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {im} (f)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>im</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {im} (f)}</annotation>
</semantics>
</math></span><img src="./d81272f637ebd155af124d31001570170a3d8112.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.671ex; height:2.843ex;" alt="{\displaystyle \operatorname {im} (f)}" loading="lazy"></span> denotes image of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle f.}</annotation>
</semantics>
</math></span><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"></span> Every one <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> maps to exactly one unique <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>
</semantics>
</math></span><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"></span> The circled parts of the axes represent domain and range sets— in accordance with the standard diagrams above</div>
</li>
</ul></div></div></div>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Bijection%2C_injection_and_surjection" title="Bijection, injection and surjection">Bijection, injection and surjection</a> – Properties of mathematical functions</li>
<li><a href="Injective_metric_space" title="Injective metric space">Injective metric space</a> – Type of metric space</li>
<li><a href="Monotonic_function" title="Monotonic function">Monotonic function</a> – Order-preserving mathematical function</li>
<li><a href="Univalent_function" title="Univalent function">Univalent function</a> – Mathematical concept</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
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</style><div class="reflist">
</div>
<div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Sometimes <i>one-one function</i>, in Indian mathematical education. <style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */
.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}
/* end https://en.wikipedia.org/ */
</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncert.nic.in/ncerts/l/lemh101.pdf">"Chapter 1:Relations and functions"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231226194119/https://ncert.nic.in/ncerts/l/lemh101.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on Dec 26, 2023 – via NCERT.</cite></span>
</li>
<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/sets/injective-surjective-bijective.html">"Injective, Surjective and Bijective"</a>. <i>Math is Fun</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-07</span></span>.</cite></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/00V5">"Section 7.3 (00V5): Injective and surjective maps of presheaves"</a>. <i>The Stacks project</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-07</span></span>.</cite></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFFarlow" class="citation web cs1"><a href="Stanley_Farlow" title="Stanley Farlow">Farlow, S. J.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191207035302/http://www.math.umaine.edu/~farlow/sec42.pdf">"Section 4.2 Injections, Surjections, and Bijections"</a> <span class="cs1-format">(PDF)</span>. <i>Mathematics & Statistics - University of Maine</i>. Archived from <a rel="nofollow" class="external text" href="http://www.math.umaine.edu/~farlow/sec42.pdf">the original</a> <span class="cs1-format">(PDF)</span> on Dec 7, 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-06</span></span>.</cite></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/46678/what-are-usual-notations-for-surjective-injective-and-bijective-functions">"What are usual notations for surjective, injective and bijective functions?"</a>. <i>Mathematics Stack Exchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-24</span></span>.</cite></span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Unlike the corresponding statement that every surjective function has a right inverse, this does not require the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, as the existence of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> is implied by the non-emptiness of the domain. However, this statement may fail in less conventional mathematics such as <a href="Constructive_mathematics" class="mw-redirect" title="Constructive mathematics">constructive mathematics</a>. In constructive mathematics, the inclusion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}\to \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{0,1\}\to \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./58e8efe0c40908e3141ec4b424ebc96e0193462d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.976ex; height:2.843ex;" alt="{\displaystyle \{0,1\}\to \mathbb {R} }" loading="lazy"></span> of the two-element set in the reals cannot have a left inverse, as it would violate <a href="Indecomposability_(constructive_mathematics)" class="mw-redirect" title="Indecomposability (constructive mathematics)">indecomposability</a>, by giving a <a href="Retract_(category_theory)" class="mw-redirect" title="Retract (category theory)">retraction</a> of the real line to the set {0,1}.</span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilliams1996" class="citation web cs1">Williams, Peter (Aug 21, 1996). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170604162511/http://www.math.csusb.edu/notes/proofs/bpf/node4.html">"Proving Functions One-to-One"</a>. <i>Department of Mathematics at CSU San Bernardino Reference Notes Page</i>. Archived from <a rel="nofollow" class="external text" href="http://www.math.csusb.edu/notes/proofs/bpf/node4.html">the original</a> on 4 June 2017.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><cite id="CITEREFBartle1976" class="citation cs2">Bartle, Robert G. (1976), <i>The Elements of Real Analysis</i> (2nd ed.), New York: <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-05464-1</bdi></cite>, p. 17 <i>ff</i>.</li>
<li><cite id="CITEREFHalmos1974" class="citation cs2"><a href="Paul_R._Halmos" class="mw-redirect" title="Paul R. Halmos">Halmos, Paul R.</a> (1974), <i><a href="Naive_Set_Theory_(book)" title="Naive Set Theory (book)">Naive Set Theory</a></i>, New York: Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90092-6</bdi></cite>, p. 38 <i>ff</i>.</li></ul>
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Injectivity" class="extiw external" title="commons:Category:Injectivity">Injectivity</a></span>.</div></div>
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<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/injective" class="extiw external" title="wiktionary:injective">injective</a></b></i> in Wiktionary, the free dictionary.</div></div>
</div>
<ul><li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/i.html">Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.</a></li>
<li><a rel="nofollow" class="external text" href="http://www.khanacademy.org/math/linear-algebra/v/surjective--onto--and-injective--one-to-one--functions">Khan Academy – Surjective (onto) and Injective (one-to-one) functions: Introduction to surjective and injective functions</a></li></ul>
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<ul><li><a href="Axiom" title="Axiom">Axiom</a>
<ul><li><a href="List_of_axioms" title="List of axioms">list</a></li></ul></li>
<li><a href="Cardinality" title="Cardinality">Cardinality</a></li>
<li><a href="First-order_logic" title="First-order logic">First-order logic</a></li>
<li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>
<li><a href="Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li>
<li><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li>
<li><a href="Information_theory" title="Information theory">Information theory</a></li>
<li><a href="Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li>
<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>
<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li>
<li><a href="Theorem" title="Theorem">Theorem</a></li>
<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li>
<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (list)<br> and <a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li>
<li><a href="Tarski's_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li>
<li><a href="Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li>
<li>Cantor's <a href="Cantor's_theorem" title="Cantor's theorem">theorem,</a> <a href="Cantor's_paradox" title="Cantor's paradox">paradox</a> and <a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li>
<li><a href="Compactness_theorem" title="Compactness theorem">Compactness</a></li>
<li><a href="Halting_problem" title="Halting problem">Halting problem</a></li>
<li><a href="Lindstr%C3%B6m's_theorem" title="Lindström's theorem">Lindström's</a></li>
<li><a href="L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li>
<li><a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional95" scope="row" class="navbox-group" style="width:1%"><a href="Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Classical_logic" title="Classical logic">Classical logic</a></li>
<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>
<li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a></li>
<li><a href="Proposition" title="Proposition">Proposition</a></li>
<li><a href="Inference" title="Inference">Inference</a></li>
<li><a href="Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li>
<li><a href="Consistency" title="Consistency">Consistency</a>
<ul><li><a href="Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li>
<li><a href="Argument" title="Argument">Argument</a></li>
<li><a href="Soundness" title="Soundness">Soundness</a></li>
<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li>
<li><a href="Syllogism" title="Syllogism">Syllogism</a></li>
<li><a href="Square_of_opposition" title="Square of opposition">Square of opposition</a></li>
<li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li>
<li><a href="Boolean_function" title="Boolean function">Boolean functions</a></li>
<li><a href="Logical_connective" title="Logical connective">Logical connectives</a></li>
<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li>
<li><a href="Propositional_formula" title="Propositional formula">Propositional formula</a></li>
<li><a href="Truth_table" title="Truth table">Truth tables</a></li>
<li><a href="Many-valued_logic" title="Many-valued logic">Many-valued logic</a>
<ul><li><a href="Three-valued_logic" title="Three-valued logic">3</a></li>
<li><a href="Finite-valued_logic" title="Finite-valued logic">finite</a></li>
<li><a href="Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="First-order_logic" title="First-order logic">First-order</a>
<ul><li><a href="List_of_first-order_theories" title="List of first-order theories"><span style="font-size: 85%;">list</span></a></li></ul></li>
<li><a href="Second-order_logic" title="Second-order logic">Second-order</a>
<ul><li><a href="Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li>
<li><a href="Higher-order_logic" title="Higher-order logic">Higher-order</a></li>
<li><a href="Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li>
<li><a href="Free_logic" title="Free logic">Free</a></li>
<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li>
<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a></li>
<li><a href="Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a>
<ul><li><a href="Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li>
<li><a href="Class_(set_theory)" title="Class (set theory)">Class</a></li>
<li>(<a href="Urelement" title="Urelement">Ur-</a>)<a href="Element_(mathematics)" title="Element (mathematics)">Element</a></li>
<li><a href="Ordinal_number" title="Ordinal number">Ordinal number</a></li>
<li><a href="Extensionality" title="Extensionality">Extensionality</a></li>
<li><a href="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li>
<li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a>
<ul><li><a href="Equivalence_relation" title="Equivalence relation">equivalence</a></li>
<li><a href="Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li>
<li>Set operations:
<ul><li><a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li>
<li><a href="Union_(set_theory)" title="Union (set theory)">union</a></li>
<li><a href="Complement_(set_theory)" title="Complement (set theory)">complement</a></li>
<li><a href="Cartesian_product" title="Cartesian product">Cartesian product</a></li>
<li><a href="Power_set" title="Power set">power set</a></li>
<li><a href="List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Countable_set" title="Countable set">Countable</a></li>
<li><a href="Uncountable_set" title="Uncountable set">Uncountable</a></li>
<li><a href="Empty_set" title="Empty set">Empty</a></li>
<li><a href="Inhabited_set" title="Inhabited set">Inhabited</a></li>
<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li>
<li><a href="Finite_set" title="Finite set">Finite</a></li>
<li><a href="Infinite_set" title="Infinite set">Infinite</a></li>
<li><a href="Transitive_set" title="Transitive set">Transitive</a></li>
<li><a href="Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li>
<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li>
<li><a href="Fuzzy_set" title="Fuzzy set">Fuzzy</a></li>
<li><a href="Universal_set" title="Universal set">Universal</a></li>
<li><a href="Universe_(mathematics)" title="Universe (mathematics)">Universe</a>
<ul><li><a href="Constructible_universe" title="Constructible universe">constructible</a></li>
<li><a href="Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li>
<li><a href="Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="Map_(mathematics)" title="Map (mathematics)">Map</a>
<ul><li><a href="Domain_of_a_function" title="Domain of a function">domain</a></li>
<li><a href="Codomain" title="Codomain">codomain</a></li>
<li><a href="Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li>
<li>/<a href="Surjective_function" title="Surjective function">Sur</a>/<a href="Bijection" title="Bijection">Bi</a>-jection</li>
<li><a href="Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li>
<li><a href="Isomorphism" title="Isomorphism">Isomorphism</a></li>
<li><a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li>
<li><a href="Enumeration" title="Enumeration">Enumeration</a></li>
<li><a href="Large_cardinal" title="Large cardinal">Large cardinal</a>
<ul><li><a href="Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li>
<li><a href="Aleph_number" title="Aleph number">Aleph number</a></li>
<li><a href="Operation_(mathematics)" title="Operation (mathematics)">Operation</a>
<ul><li><a href="Binary_operation" title="Binary operation">binary</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a>
<ul><li><a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li>
<li><a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li>
<li><a href="General_set_theory" title="General set theory">General</a></li>
<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li>
<li><a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li>
<li><a href="Naive_set_theory" title="Naive set theory">Naive</a></li>
<li><a href="New_Foundations" title="New Foundations">New Foundations</a></li>
<li><a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li>
<li><a href="Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li>
<li><a href="Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li>
<li><a href="Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Formal_system" title="Formal system">Formal systems</a> (<a href="List_of_formal_systems" title="List of formal systems"><span style="font-size: 85%;">list</span></a>),<br><a href="Formal_language" title="Formal language">language</a> and <a href="Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li>
<li><a href="Arity" title="Arity">Arity</a></li>
<li><a href="Automata_theory" title="Automata theory">Automata</a></li>
<li><a href="Axiom_schema" title="Axiom schema">Axiom schema</a></li>
<li><a href="Expression_(mathematics)" title="Expression (mathematics)">Expression</a>
<ul><li><a href="Ground_expression" title="Ground expression">ground</a></li></ul></li>
<li><a href="Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a>
<ul><li><a href="Extension_by_definitions" class="mw-redirect" title="Extension by definitions">by definition</a></li>
<li><a href="Conservative_extension" title="Conservative extension">conservative</a></li></ul></li>
<li><a href="Finitary_relation" title="Finitary relation">Relation</a></li>
<li><a href="Formation_rule" title="Formation rule">Formation rule</a></li>
<li><a href="Formal_grammar" title="Formal grammar">Grammar</a></li>
<li><a href="Well-formed_formula" title="Well-formed formula">Formula</a>
<ul><li><a href="Atomic_formula" title="Atomic formula">atomic</a></li>
<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li>
<li><a href="Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li>
<li><a href="Open_formula" title="Open formula">open</a></li></ul></li>
<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li>
<li><a href="Formal_language" title="Formal language">Language</a></li>
<li><a href="Metalanguage" title="Metalanguage">Metalanguage</a></li>
<li><a href="Logical_connective" title="Logical connective">Logical connective</a>
<ul><li><a href="Negation" title="Negation">¬</a></li>
<li><a href="Logical_disjunction" title="Logical disjunction">∨</a></li>
<li><a href="Logical_conjunction" title="Logical conjunction">∧</a></li>
<li><a href="Material_conditional" title="Material conditional">→</a></li>
<li><a href="Logical_biconditional" title="Logical biconditional">↔</a></li>
<li><a href="Logical_equality" title="Logical equality">=</a></li></ul></li>
<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a>
<ul><li><a href="Functional_predicate" title="Functional predicate">functional</a></li>
<li><a href="Predicate_variable" title="Predicate variable">variable</a></li>
<li><a href="Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li>
<li><a href="Formal_proof" title="Formal proof">Proof</a></li>
<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a>
<ul><li><a href="Existential_quantification" title="Existential quantification">∃</a></li>
<li><a href="Uniqueness_quantification" title="Uniqueness quantification">!</a></li>
<li><a href="Universal_quantification" title="Universal quantification">∀</a></li>
<li><a href="Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li>
<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a>
<ul><li><a href="Atomic_sentence" title="Atomic sentence">atomic</a></li>
<li><a href="Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li>
<li><a href="Signature_(logic)" title="Signature (logic)">Signature</a></li>
<li><a href="String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li>
<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>
<li><a href="Symbol_(formal)" title="Symbol (formal)">Symbol</a>
<ul><li><a href="Uninterpreted_function" title="Uninterpreted function">function</a></li>
<li><a href="Logical_constant" title="Logical constant">logical/constant</a></li>
<li><a href="Non-logical_symbol" title="Non-logical symbol">non-logical</a></li>
<li><a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li>
<li><a href="Term_(logic)" title="Term (logic)">Term</a></li>
<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a>
<ul><li><a href="List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size: 85%;">list</span></a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="Axiomatic_system" title="Axiomatic system">axiomatic<br>systems</a> <span style="font-size: 85%;">(<a href="List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li>of <a href="True_arithmetic" title="True arithmetic">arithmetic</a>:
<ul><li><a href="Peano_axioms" title="Peano axioms">Peano</a></li>
<li><a href="Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li>
<li><a href="Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li>
<li><a href="Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li>
<li><a href="Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li>
<li><a href="Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li>
<li>of the <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a>
<ul><li><a href="Tarski's_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li>
<li>of <a href="Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a>
<ul><li><a href="Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li>
<li><a href="Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li>
<li>of <a href="Foundations_of_geometry" title="Foundations of geometry">geometry</a>:
<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a>:
<ul><li><a href="Euclid's_Elements" title="Euclid's Elements"><i>Elements</i></a></li>
<li><a href="Hilbert's_axioms" title="Hilbert's axioms">Hilbert's</a></li>
<li><a href="Tarski's_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li>
<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul>
<ul><li><i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>
<li><a href="Natural_deduction" title="Natural deduction">Natural deduction</a></li>
<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>
<li><a href="Rule_of_inference" title="Rule of inference">Rule of inference</a></li>
<li><a href="Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li>
<li><a href="Theorem" title="Theorem">Theorem</a></li>
<li><a href="Formal_system" title="Formal system">Systems</a>
<ul><li><a href="Axiomatic_system" title="Axiomatic system">axiomatic</a></li>
<li><a href="Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li>
<li><a href="Hilbert_system" title="Hilbert system">Hilbert</a>
<ul><li><a href="List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li>
<li><a href="Complete_theory" title="Complete theory">Complete theory</a></li>
<li><a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li>
<li><a href="Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li>
<li><a href="Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li>
<li><a href="Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li>
<li><a href="Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a>
<ul><li><a href="Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li>
<li><a href="Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li>
<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a>
<ul><li><a href="Elementary_equivalence" title="Elementary equivalence">equivalence</a></li>
<li><a href="Finite_model_theory" title="Finite model theory">finite</a></li>
<li><a href="Saturated_model" title="Saturated model">saturated</a></li>
<li><a href="Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li>
<li><a href="Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li>
<li><a href="Non-standard_model" title="Non-standard model">Non-standard model</a>
<ul><li><a href="Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li>
<li><a href="Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a>
<ul><li><a href="Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li>
<li><a href="Categorical_theory" title="Categorical theory">Categorical theory</a></li>
<li><a href="Model_complete_theory" title="Model complete theory">Model complete theory</a></li>
<li><a href="Satisfiability" title="Satisfiability">Satisfiability</a></li>
<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>
<li><a href="Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li>
<li><a href="Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a>
<ul><li><a href="Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li>
<li><a href="Tarski's_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li>
<li><a href="Kripke's_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li>
<li><a href="T-schema" title="T-schema">T-schema</a></li>
<li><a href="Transfer_principle" title="Transfer principle">Transfer principle</a></li>
<li><a href="Truth_predicate" title="Truth predicate">Truth predicate</a></li>
<li><a href="Truth_value" title="Truth value">Truth value</a></li>
<li><a href="Type_(model_theory)" title="Type (model theory)">Type</a></li>
<li><a href="Ultraproduct" title="Ultraproduct">Ultraproduct</a></li>
<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Church_encoding" title="Church encoding">Church encoding</a></li>
<li><a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li>
<li><a href="Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li>
<li><a href="Computable_function" title="Computable function">Computable function</a></li>
<li><a href="Computable_set" title="Computable set">Computable set</a></li>
<li><a href="Decision_problem" title="Decision problem">Decision problem</a>
<ul><li><a href="Decidability_(logic)" title="Decidability (logic)">decidable</a></li>
<li><a href="Undecidable_problem" title="Undecidable problem">undecidable</a></li>
<li><a href="P_(complexity)" title="P (complexity)">P</a></li>
<li><a href="NP_(complexity)" title="NP (complexity)">NP</a></li>
<li><a href="P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li>
<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>
<li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>
<li><a href="Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li>
<li><a href="Recursion" title="Recursion">Recursion</a></li>
<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li>
<li><a href="Turing_machine" title="Turing machine">Turing machine</a></li>
<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Abstract_logic" title="Abstract logic">Abstract logic</a></li>
<li><a href="Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li>
<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>
<li><a href="Category_theory" title="Category theory">Category theory</a></li>
<li><a href="Concrete_category" title="Concrete category">Concrete</a>/<a href="Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li>
<li><a href="Category_of_sets" title="Category of sets">Category of sets</a></li>
<li><a href="History_of_logic" title="History of logic">History of logic</a></li>
<li><a href="History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a>
<ul><li><a href="Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li>
<li><a href="Logicism" title="Logicism">Logicism</a></li>
<li><a href="Mathematical_object" title="Mathematical object">Mathematical object</a></li>
<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li>
<li><a href="Supertask" title="Supertask">Supertask</a></li></ul>
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