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Set-builder notation

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<annotation encoding="application/x-tex">{\displaystyle \{n\mid \exists k\in \mathbb {Z} ,n=2k\}}</annotation>

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</math><img src="./3d08d5bbe9574a241428e503b8733c3d7366e413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.452ex; height:2.843ex;" alt="{\displaystyle \{n\mid \exists k\in \mathbb {Z} ,n=2k\}}" loading="lazy">

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<div style="padding-bottom: 0; padding-top: 0.5em"><cite class="left-aligned" style="">The set of all <a href="Even_integer" class="mw-redirect" title="Even integer">even integers</a>, expressed in set-builder notation.</cite></div>

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In <a href="Mathematics" title="Mathematics">mathematics</a> and more specifically in <a href="Set_theory" title="Set theory">set theory</a>, set-builder notation is a <a href="Mathematical_notation" title="Mathematical notation">notation</a> for specifying a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> by a property that characterizes its members.<a href="#cite_note-1">[1]</a>

Specifying sets by member properties is allowed by the <a href="Axiom_schema_of_specification" title="Axiom schema of specification">axiom schema of specification</a>. This is also known as set comprehension and set abstraction.

<div class="mw-heading mw-heading2"><h2 id="Sets_defined_by_a_predicate">Sets defined by a predicate</h2></div>

Set-builder notation can be used to describe a set that is defined by a <a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">predicate</a>, that is, a logical formula that evaluates to true for an element of the set, and false otherwise.<a href="#cite_note-2">[2]</a> In this form, set-builder notation has three parts: a variable, a <a href="Colon_(punctuation)" title="Colon (punctuation)">colon</a> or <a href="Vertical_bar" title="Vertical bar">vertical bar</a> separator, and a predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets:

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<annotation encoding="application/x-tex">{\displaystyle \{x\mid \Phi (x)\}}</annotation>

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</math><img src="./fc7f6ac571aefc622d3792ddebb869257104f136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.409ex; height:2.843ex;" alt="{\displaystyle \{x\mid \Phi (x)\}}" loading="lazy"></dd></dl>

or

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<annotation encoding="application/x-tex">{\displaystyle \{x:\Phi (x)\}.}</annotation>

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</math><img src="./38f462eb953a6cca7a9c7a4ae603102a188dd408.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.056ex; height:2.843ex;" alt="{\displaystyle \{x:\Phi (x)\}.}" loading="lazy"></dd></dl>

The vertical bar (or colon) is a separator that can be read as "such that", "for which", or "with the property that". The formula Φ(x) is said to be the rule or the predicate. All values of x for which the predicate holds (is true) belong to the set being defined. All values of x for which the predicate does not hold do not belong to the set. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\mid \Phi (x)\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x\mid \Phi (x)\}}</annotation>

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</math><img src="./fc7f6ac571aefc622d3792ddebb869257104f136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.409ex; height:2.843ex;" alt="{\displaystyle \{x\mid \Phi (x)\}}" loading="lazy"> is the set of all values of x that satisfy the formula Φ.<a href="#cite_note-3">[3]</a> It may be the <a href="Empty_set" title="Empty set">empty set</a>, if no value of x satisfies the formula.

<div class="mw-heading mw-heading3"><h3 id="Specifying_the_domain">Specifying the domain</h3></div>

A domain E can appear on the left of the vertical bar:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle \{x\in E\mid \Phi (x)\},}</annotation>

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</math><img src="./3b3d31b7fe6a9a12eb0a412f251a9624cb42d981.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.672ex; height:2.843ex;" alt="{\displaystyle \{x\in E\mid \Phi (x)\},}" loading="lazy"></dd></dl>

or by adjoining it to the predicate:

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<annotation encoding="application/x-tex">{\displaystyle \{x\mid x\in E{\text{ and }}\Phi (x)\}\quad {\text{or}}\quad \{x\mid x\in E\land \Phi (x)\}.}</annotation>

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</math><img src="./49a118e0efa5fdd6ff35fe0b617f6ab38e11a4ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.567ex; height:2.843ex;" alt="{\displaystyle \{x\mid x\in E{\text{ and }}\Phi (x)\}\quad {\text{or}}\quad \{x\mid x\in E\land \Phi (x)\}.}" loading="lazy"></dd></dl>

The ∈ symbol here denotes <a href="Set_membership" class="mw-redirect" title="Set membership">set membership</a>, while the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }">

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<annotation encoding="application/x-tex">{\displaystyle \land }</annotation>

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</math><img src="./d6823e5a222eb3ca49672818ac3d13ec607052c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }" loading="lazy"> symbol denotes the logical "and" operator, known as <a href="Logical_conjunction" title="Logical conjunction">logical conjunction</a>. This notation represents the set of all values of x that belong to some given set E for which the predicate is true (see "<a href="#Set_existence_axiom">Set existence axiom</a>" below). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation>

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</math><img src="./79e4f01c93494fbb5dcd75761f4468121b00b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="{\displaystyle \Phi (x)}" loading="lazy"> is a conjunction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{1}(x)\land \Phi _{2}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{1}(x)\land \Phi _{2}(x)}</annotation>

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</math><img src="./a292f98818d148527a19e4f3e6120dd4170eaf9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.325ex; height:2.843ex;" alt="{\displaystyle \Phi _{1}(x)\land \Phi _{2}(x)}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in E\mid \Phi (x)\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x\in E\mid \Phi (x)\}}</annotation>

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</math><img src="./56f8838e80a235f133c751bc4c3b26ececa685ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.025ex; height:2.843ex;" alt="{\displaystyle \{x\in E\mid \Phi (x)\}}" loading="lazy"> is sometimes written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in E\mid \Phi _{1}(x),\Phi _{2}(x)\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x\in E\mid \Phi _{1}(x),\Phi _{2}(x)\}}</annotation>

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</math><img src="./8a9382804c62441be0a6ae7202c9da7f0e49f6ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.985ex; height:2.843ex;" alt="{\displaystyle \{x\in E\mid \Phi _{1}(x),\Phi _{2}(x)\}}" loading="lazy">, using a comma instead of the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }">

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<annotation encoding="application/x-tex">{\displaystyle \land }</annotation>

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</math><img src="./d6823e5a222eb3ca49672818ac3d13ec607052c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }" loading="lazy">.

In general, it is not a good idea to consider sets without defining a <a href="Domain_of_discourse" title="Domain of discourse">domain of discourse</a>, as this would represent the <a href="Subset" title="Subset">subset</a> of all possible things that may exist for which the predicate is true. This can easily lead to contradictions and paradoxes. For example, <a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a> shows that the expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x~|~x\not \in x\},}">

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<annotation encoding="application/x-tex">{\displaystyle \{x~|~x\not \in x\},}</annotation>

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</math><img src="./f53b7fc3cd6fa07ae176f8e425bd9ac2364acfa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.61ex; height:2.843ex;" alt="{\displaystyle \{x~|~x\not \in x\},}" loading="lazy"> although seemingly well formed as a set builder expression, cannot define a set without producing a contradiction.<a href="#cite_note-5">[5]</a>

In cases where the set E is clear from context, it may be not explicitly specified. It is common in the literature for an author to state the domain ahead of time, and then not specify it in the set-builder notation. For example, an author may say something such as, "Unless otherwise stated, variables are to be taken to be natural numbers," though in less formal contexts where the domain can be assumed, a written mention is often unnecessary.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

The following examples illustrate particular sets defined by set-builder notation via predicates. In each case, the domain is specified on the left side of the vertical bar, while the rule is specified on the right side.

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<annotation encoding="application/x-tex">{\displaystyle \{x\in \mathbb {R} \mid x>0\}}</annotation>

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</math><img src="./91e0a3d952ba4157408ad6e72933404845089761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.701ex; height:2.843ex;" alt="{\displaystyle \{x\in \mathbb {R} \mid x>0\}}" loading="lazy"> is the set of all strictly <a href="Positive_number" class="mw-redirect" title="Positive number">positive</a> <a href="Real_number" title="Real number">real numbers</a>, which can be written in <a href="Interval_notation" class="mw-redirect" title="Interval notation">interval notation</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle (0,\infty )}</annotation>

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</math><img src="./da17102e4ed0886686094ee531df040d2e86352a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (0,\infty )}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle \{x\in \mathbb {R} \mid |x|=1\}}</annotation>

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</math><img src="./739bee31757c29b0a430e82d7274f55b962d4e68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.995ex; height:2.843ex;" alt="{\displaystyle \{x\in \mathbb {R} \mid |x|=1\}}" loading="lazy"> is the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-1,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{-1,1\}}</annotation>

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</math><img src="./c0ffb8c7a09dad8456eee3669ee9a7e462fe3c34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.492ex; height:2.843ex;" alt="{\displaystyle \{-1,1\}}" loading="lazy">. This set can also be defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}}</annotation>

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</math><img src="./775f463b4267457fac55b577395b6b211dc9fe69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.755ex; height:3.176ex;" alt="{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}}" loading="lazy">; see <a href="#Equivalent_predicates_yield_equal_sets">equivalent predicates yield equal sets</a> below.</li>

<li>For each integer m, we can define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{m}=\{x\in \mathbb {Z} \mid x\geq m\}=\{m,m+1,m+2,\ldots \}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{m}=\{x\in \mathbb {Z} \mid x\geq m\}=\{m,m+1,m+2,\ldots \}}</annotation>

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</math><img src="./3e6b318b1769ab0c58bbd749afa1a34882de85fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.427ex; height:2.843ex;" alt="{\displaystyle G_{m}=\{x\in \mathbb {Z} \mid x\geq m\}=\{m,m+1,m+2,\ldots \}}" loading="lazy">. As an example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{3}=\{x\in \mathbb {Z} \mid x\geq 3\}=\{3,4,5,\ldots \}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{3}=\{x\in \mathbb {Z} \mid x\geq 3\}=\{3,4,5,\ldots \}}</annotation>

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</math><img src="./0cf668c1e0463567c2c2578662c1e7706f2625e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.289ex; height:2.843ex;" alt="{\displaystyle G_{3}=\{x\in \mathbb {Z} \mid x\geq 3\}=\{3,4,5,\ldots \}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{-2}=\{-2,-1,0,\ldots \}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{-2}=\{-2,-1,0,\ldots \}}</annotation>

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</math><img src="./880004f1c2bc687a3b1b04301f925d7ca64c5676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.512ex; height:2.843ex;" alt="{\displaystyle G_{-2}=\{-2,-1,0,\ldots \}}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle \{(x,y)\in \mathbb {R} \times \mathbb {R} \mid 0<y<f(x)\}}</annotation>

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</math><img src="./f875581ef48fc23f6041506cad847cda66a04180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.56ex; height:2.843ex;" alt="{\displaystyle \{(x,y)\in \mathbb {R} \times \mathbb {R} \mid 0<y<f(x)\}}" loading="lazy"> is the set of pairs of real numbers such that y is greater than 0 and less than f(x), for a given <a href="Function_(mathematics)" title="Function (mathematics)">function</a> f. Here the <a href="Cartesian_product" title="Cartesian product">cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \times \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \mathbb {R} }</annotation>

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</math><img src="./eb99f01c438a62e4ac5af8cff4eb402739ed67a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \times \mathbb {R} }" loading="lazy"> denotes the set of ordered pairs of real numbers.</li>

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<annotation encoding="application/x-tex">{\displaystyle \{n\in \mathbb {N} \mid (\exists k)[k\in \mathbb {N} \land n=2k]\}}</annotation>

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</math><img src="./5a46338b7739df2ad6f24c515613810639c73439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.962ex; height:2.843ex;" alt="{\displaystyle \{n\in \mathbb {N} \mid (\exists k)[k\in \mathbb {N} \land n=2k]\}}" loading="lazy"> is the set of all <a href="Even_number" class="mw-redirect" title="Even number">even</a> <a href="Natural_number" title="Natural number">natural numbers</a>. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }">

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<annotation encoding="application/x-tex">{\displaystyle \land }</annotation>

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</math><img src="./d6823e5a222eb3ca49672818ac3d13ec607052c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }" loading="lazy"> sign stands for "and", which is known as <a href="Logical_conjunction" title="Logical conjunction">logical conjunction</a>. The ∃ sign stands for "there exists", which is known as <a href="Existential_quantification" title="Existential quantification">existential quantification</a>. So for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists x)P(x)}">

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<annotation encoding="application/x-tex">{\displaystyle (\exists x)P(x)}</annotation>

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</math><img src="./8b97f48e1a2037bb39ad6f54fb60acc01c62c2c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.316ex; height:2.843ex;" alt="{\displaystyle (\exists x)P(x)}" loading="lazy"> is read as "there exists an x such that P(x)".</li>

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<annotation encoding="application/x-tex">{\displaystyle \{n\mid (\exists k\in \mathbb {N} )[n=2k]\}}</annotation>

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</math><img src="./701ae3a0b2195e3749d19cdc48f21418042dbd00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.649ex; height:2.843ex;" alt="{\displaystyle \{n\mid (\exists k\in \mathbb {N} )[n=2k]\}}" loading="lazy"> is a notational variant for the same set of even natural numbers. It is not necessary to specify that n is a natural number, as this is implied by the formula on the right.</li>

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<annotation encoding="application/x-tex">{\displaystyle \{a\in \mathbb {R} \mid (\exists p\in \mathbb {Z} )(\exists q\in \mathbb {Z} )[q\not =0\land aq=p]\}}</annotation>

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</math><img src="./4aaa130fdd174335a4c527fe7e7aa62e723642c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.009ex; height:2.843ex;" alt="{\displaystyle \{a\in \mathbb {R} \mid (\exists p\in \mathbb {Z} )(\exists q\in \mathbb {Z} )[q\not =0\land aq=p]\}}" loading="lazy"> is the set of <a href="Rational_number" title="Rational number">rational numbers</a>; that is, real numbers that can be written as the ratio of two <a href="Integer" title="Integer">integers</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="More_complex_expressions_on_the_left_side_of_the_notation">More complex expressions on the left side of the notation</h2></div>

An extension of set-builder notation replaces the single variable x with an <a href="Expression_(mathematics)" title="Expression (mathematics)">expression</a>. So instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\mid \Phi (x)\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x\mid \Phi (x)\}}</annotation>

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</math><img src="./fc7f6ac571aefc622d3792ddebb869257104f136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.409ex; height:2.843ex;" alt="{\displaystyle \{x\mid \Phi (x)\}}" loading="lazy">, we may have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f(x)\mid \Phi (x)\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{f(x)\mid \Phi (x)\},}</annotation>

</semantics>

</math><img src="./48aa87f275c9701a8584a05ce832acc150e8eb52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.144ex; height:2.843ex;" alt="{\displaystyle \{f(x)\mid \Phi (x)\},}" loading="lazy"> which should be read

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{f(x)\mid \Phi (x)\}=\{y\mid \exists x(y=f(x)\wedge \Phi (x))\}}</annotation>

</semantics>

</math><img src="./8037be64f0b135ade9480f0ea17af58b1e88df27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.515ex; height:2.843ex;" alt="{\displaystyle \{f(x)\mid \Phi (x)\}=\{y\mid \exists x(y=f(x)\wedge \Phi (x))\}}" loading="lazy">.</dd></dl>

For example:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{2n\mid n\in \mathbb {N} \}}</annotation>

</semantics>

</math><img src="./8ce92765655dbc7b1b726f07683c5448f0b43c7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.733ex; height:2.843ex;" alt="{\displaystyle \{2n\mid n\in \mathbb {N} \}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

</semantics>

</math><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"> is the set of all natural numbers, is the set of all even natural numbers.</li>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{p/q\mid p,q\in \mathbb {Z} ,q\not =0\}}</annotation>

</semantics>

</math><img src="./b5ac1cae555e4634488a2b19bf3feafdf30e0df5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.692ex; height:2.843ex;" alt="{\displaystyle \{p/q\mid p,q\in \mathbb {Z} ,q\not =0\}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> is the set of all integers, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ,}</annotation>

</semantics>

</math><img src="./91185244fbdded6ea99a5e9e6603299128b10928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,}" loading="lazy"> the set of all rational numbers.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}}</annotation>

</semantics>

</math><img src="./778c2185ece69ae5912d3ec1c41c0a9371025bbd.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 \{2t+1\mid t\in \mathbb {Z} \}}" loading="lazy"> is the set of odd integers.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(t,2t+1)\mid t\in \mathbb {Z} \}}</annotation>

</semantics>

</math><img src="./62feaf63e58cd386b828b28eccd58ce2dd0bb067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.18ex; height:2.843ex;" alt="{\displaystyle \{(t,2t+1)\mid t\in \mathbb {Z} \}}" loading="lazy"> creates a set of pairs, where each pair puts an integer into correspondence with an odd integer.</li></ul>

When inverse functions can be explicitly stated, the expression on the left can be eliminated through simple substitution. Consider the example set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}}</annotation>

</semantics>

</math><img src="./778c2185ece69ae5912d3ec1c41c0a9371025bbd.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 \{2t+1\mid t\in \mathbb {Z} \}}" loading="lazy">. Make the substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=2t+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=2t+1}</annotation>

</semantics>

</math><img src="./f8edda6a9a8c3731e56db7676aa4991c02f2a832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.433ex; height:2.343ex;" alt="{\displaystyle u=2t+1}" loading="lazy">, which is to say <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=(u-1)/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=(u-1)/2}</annotation>

</semantics>

</math><img src="./5b2dd0563394c8180c8abcd7106f2b5b2f3609f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.405ex; height:2.843ex;" alt="{\displaystyle t=(u-1)/2}" loading="lazy">, then replace t in the set builder notation to find

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}=\{u\mid (u-1)/2\in \mathbb {Z} \}.}</annotation>

</semantics>

</math><img src="./326b8b16c6c6e171ceb9999f95d8a4bebb9a4433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.692ex; height:2.843ex;" alt="{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}=\{u\mid (u-1)/2\in \mathbb {Z} \}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Equivalent_predicates_yield_equal_sets">Equivalent predicates yield equal sets</h2></div>

Two sets are equal if and only if they have the same elements. Sets defined by set builder notation are equal if and only if their set builder rules, including the domain specifiers, are equivalent. That is

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\in A\mid P(x)\}=\{x\in B\mid Q(x)\}}</annotation>

</semantics>

</math><img src="./8b98c815026a915a2104deed3b354a242833ea2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.332ex; height:2.843ex;" alt="{\displaystyle \{x\in A\mid P(x)\}=\{x\in B\mid Q(x)\}}" loading="lazy"></dd></dl>

if and only if

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\forall t)[(t\in A\land P(t))\Leftrightarrow (t\in B\land Q(t))]}</annotation>

</semantics>

</math><img src="./658c37b78c51b9bbf5310d58c1497a0ebfaf95e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.382ex; height:2.843ex;" alt="{\displaystyle (\forall t)[(t\in A\land P(t))\Leftrightarrow (t\in B\land Q(t))]}" loading="lazy">.</dd></dl>

Therefore, in order to prove the equality of two sets defined by set builder notation, it suffices to prove the equivalence of their predicates, including the domain qualifiers.

For example,

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}=\{x\in \mathbb {Q} \mid |x|=1\}}</annotation>

</semantics>

</math><img src="./e092f38e7fcabebc72f44fce0ffa388b5decba5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.979ex; height:3.176ex;" alt="{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}=\{x\in \mathbb {Q} \mid |x|=1\}}" loading="lazy"></dd></dl>

because the two rule predicates are logically equivalent:

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x\in \mathbb {R} \land x^{2}=1)\Leftrightarrow (x\in \mathbb {Q} \land |x|=1).}</annotation>

</semantics>

</math><img src="./a1bcf70ce4b0cc8f86e57d13c716490d68f8003a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.401ex; height:3.176ex;" alt="{\displaystyle (x\in \mathbb {R} \land x^{2}=1)\Leftrightarrow (x\in \mathbb {Q} \land |x|=1).}" loading="lazy"></dd></dl>

This equivalence holds because, for any real number x, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}=1}</annotation>

</semantics>

</math><img src="./51e28d7561fa4fe556f438b88380ef98c5631cf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.645ex; height:2.676ex;" alt="{\displaystyle x^{2}=1}" loading="lazy"> if and only if x is a rational number with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|=1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x|=1}</annotation>

</semantics>

</math><img src="./a6af0ada8d94dff5f2130cc4ea61c6559294b85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |x|=1}" loading="lazy">. In particular, both sets are equal to the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-1,1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{-1,1\}}</annotation>

</semantics>

</math><img src="./c0ffb8c7a09dad8456eee3669ee9a7e462fe3c34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.492ex; height:2.843ex;" alt="{\displaystyle \{-1,1\}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Set_existence_axiom">Set existence axiom</h2></div>

In many formal set theories, such as <a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>, set builder notation is not part of the formal syntax of the theory. Instead, there is a <a href="Axiom_of_comprehension" class="mw-redirect" title="Axiom of comprehension">set existence axiom scheme</a>, which states that if E is a set and Φ(x) is a formula in the language of set theory, then there is a set Y whose members are exactly the elements of E that satisfy Φ:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\forall E)(\exists Y)(\forall x)[x\in Y\Leftrightarrow x\in E\land \Phi (x)].}</annotation>

</semantics>

</math><img src="./c5bfce627bbb50ca3adaad3102bc95e34e9d9862.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.028ex; height:2.843ex;" alt="{\displaystyle (\forall E)(\exists Y)(\forall x)[x\in Y\Leftrightarrow x\in E\land \Phi (x)].}" loading="lazy"></dd></dl>

The set Y obtained from this axiom is exactly the set described in set builder notation as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in E\mid \Phi (x)\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\in E\mid \Phi (x)\}}</annotation>

</semantics>

</math><img src="./56f8838e80a235f133c751bc4c3b26ececa685ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.025ex; height:2.843ex;" alt="{\displaystyle \{x\in E\mid \Phi (x)\}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="In_programming_languages">In programming languages</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="List_comprehension" title="List comprehension">List comprehension</a></div>

A similar notation available in a number of <a href="Programming_languages" class="mw-redirect" title="Programming languages">programming languages</a> (notably <a href="Python_(programming_language)" title="Python (programming language)">Python</a> and <a href="Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a>) is the <a href="List_comprehension" title="List comprehension">list comprehension</a>, which combines <a href="Map_(higher-order_function)" title="Map (higher-order function)">map</a> and <a href="Filter_(higher-order_function)" title="Filter (higher-order function)">filter</a> operations over one or more <a href="List_(computing)" class="mw-redirect" title="List (computing)">lists</a>.

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</style>

In Python, the set-builder's braces are replaced with square brackets, parentheses, or curly braces, giving list, <a href="Generator_(computer_science)" class="mw-redirect" title="Generator (computer science)">generator</a>, and set objects, respectively. Python uses an English-based syntax. Haskell replaces the set-builder's braces with square brackets and uses symbols, including the standard set-builder vertical bar.

The same can be achieved in <a href="Scala_(programming_language)" title="Scala (programming language)">Scala</a> using Sequence Comprehensions, where the "for" keyword returns a list of the yielded variables using the "yield" keyword.<a href="#cite_note-6">[6]</a>

Consider these set-builder notation examples in some programming languages:

<th>Example 1</th>

<th>Example 2

</th></tr>

<th>Set-builder

</th>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{l\ |\ l\in L\}}</annotation>

</semantics>

</math><img src="./6d05d0641bb3cc7a11c0cedf017c032eeef0291d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.943ex; height:2.843ex;" alt="{\displaystyle \{l\ |\ l\in L\}}" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(k,x)\ |\ k\in K\wedge x\in X\wedge P(x)\}}</annotation>

</semantics>

</math><img src="./e2a774243ad8269f996d957f0be4b09dda8dc7b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.835ex; height:2.843ex;" alt="{\displaystyle \{(k,x)\ |\ k\in K\wedge x\in X\wedge P(x)\}}" loading="lazy">

</td></tr>

<tr>

<th scope="row"><a href="Python_(programming_language)" title="Python (programming language)">Python</a>

</th>

<td><div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>{l for l in L}

</pre></div></td>

<td><div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>{(k, x) for k in K for x in X if P(x)}

</pre></div>

</td></tr>

<tr>

<th scope="row"><a href="Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a>

</th>

<td><div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre>[l | l <- ls]

</pre></div></td>

<td><div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre>[(k, x) | k <- ks, x <- xs, p x]

</pre></div>

</td></tr>

<tr>

<th scope="row"><a href="Scala_(programming_language)" title="Scala (programming language)">Scala</a>

</th>

<td><div class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><pre>for (l <- L) yield l

</pre></div></td>

<td><div class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><pre>for (k <- K; x <- X if P(x)) yield (k,x)

</pre></div>

</td></tr>

<tr>

</th>

<td><div class="mw-highlight mw-highlight-lang-csharp mw-content-ltr" dir="ltr"><pre>from l in L select l

</pre></div></td>

<td><div class="mw-highlight mw-highlight-lang-csharp mw-content-ltr" dir="ltr"><pre>from k in K from x in X where P(x) select (k,x)

</pre></div>

</td></tr>

<tr>

</th>

<td><div class="mw-highlight mw-highlight-lang-sql mw-content-ltr" dir="ltr"><pre>SELECT l FROM L_set

</pre></div></td>

<td><div class="mw-highlight mw-highlight-lang-sql mw-content-ltr" dir="ltr"><pre> SELECT k, x FROM K_set, X_set WHERE P(x)

</pre></div>

</td></tr>

<tr>

<th scope="row"><a href="Prolog" title="Prolog">Prolog</a>

</th>

<td><div class="mw-highlight mw-highlight-lang-prolog mw-content-ltr" dir="ltr"><pre>setof(L,member(L,Ls),Result)

</pre></div></td>

<td><div class="mw-highlight mw-highlight-lang-prolog mw-content-ltr" dir="ltr"><pre>setof((K,X),(member(K,Ks),member(X,Xs),call(P,X)),Result)

</pre></div>

</td></tr>

<tr>

<th><a href="Erlang_(programming_language)" title="Erlang (programming language)">Erlang</a>

</th>

<td><div class="mw-highlight mw-highlight-lang-erlang mw-content-ltr" dir="ltr"><pre>[L || L <- Ls]

</pre></div>

</td>

<td><div class="mw-highlight mw-highlight-lang-erlang mw-content-ltr" dir="ltr"><pre>[{K,X} || K <- Ks, X <- Xs, p(X)]

</pre></div>

</td></tr>

<tr>

<th><a href="Julia_(programming_language)" title="Julia (programming language)">Julia</a>

</th>

<td><div class="mw-highlight mw-highlight-lang-julia mw-content-ltr" dir="ltr"><pre>[l for l ∈ L]

</pre></div>

</td>

<td><div class="mw-highlight mw-highlight-lang-julia mw-content-ltr" dir="ltr"><pre>[(k, x) for k ∈ K for x ∈ X if P(x)]

</pre></div>

</td></tr>

<tr>

<th><a href="Wolfram_Mathematica" class="mw-redirect" title="Wolfram Mathematica">Mathematica</a>

</th>

<td><div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre>(l |-> l) /@ L

</pre></div>

</td>

<td><div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre>Cases[Tuples[{K, X}], {k_, x_} /; P[x]]

</pre></div>

</td></tr></tbody></table>

The set builder notation and list comprehension notation are both instances of a more general notation known as monad comprehensions, which permits map/filter-like operations over any <a href="Monad_(functional_programming)" title="Monad (functional programming)">monad</a> with a <a href="Zero_element" title="Zero element">zero element</a>.

<ul><li><a href="Glossary_of_set_theory" title="Glossary of set theory">Glossary of set theory</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFRosen2007" class="citation book cs1">Rosen, Kenneth (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZYZfQgAACAAJ&pg=PA111">Discrete Mathematics and its Applications</a> (6th ed.). New York, NY: McGraw-Hill. pp. 111–112. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-288008-3</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Michael J Cullinan, 2012, A Transition to Mathematics with Proofs, Jones & Bartlett, pp. 44ff.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Set.html">"Set"</a>. mathworld.wolfram.com. Retrieved 20 August 2020.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/sets/set-builder-notation.html">"Set-Builder Notation"</a>. mathsisfun.com. Retrieved 20 August 2020.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFIrvineDeutsch2016" class="citation encyclopaedia cs1">Irvine, Andrew David; Deutsch, Harry (9 October 2016) [1995]. <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/russell-paradox/">"Russell's Paradox"</a>. Stanford Encyclopedia of Philosophy. Retrieved 6 August 2017.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://docs.scala-lang.org/tutorials/tour/sequence-comprehensions.html">"Sequence Comprehensions"</a>. Scala. Retrieved 6 August 2017.</cite>

</li>

</ol></div></div>

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</style><div id="Set_theory409" style="font-size:114%;margin:0 4em"><a href="Set_theory" title="Set theory">Set theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</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="Set_(mathematics)" title="Set (mathematics)">Set (mathematics)</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><a href="Venn_diagram" title="Venn diagram"></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Axiom" title="Axiom">Axioms</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="Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li>

<li><a href="Axiom_of_choice" title="Axiom of choice">Choice</a>

<ul><li><a href="Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li>

<li><a href="Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li>

<li><a href="Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li>

<li><a href="Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li>

<li><a href="Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a>

<ul><li><a href="Axiom_of_projective_determinacy" class="mw-redirect" title="Axiom of projective determinacy">projective</a></li></ul></li>

<li><a href="Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li>

<li><a href="Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li>

<li><a href="Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li>

<li><a href="Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li>

<li><a href="Axiom_of_power_set" title="Axiom of power set">Power set</a></li>

<li><a href="Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li>

<li><a href="Axiom_of_union" title="Axiom of union">Union</a></li>

<li><a href="Martin's_axiom" title="Martin's axiom">Martin's axiom</a></li></ul>

<ul><li><a href="Axiom_schema" title="Axiom schema">Axiom schema</a>

<ul><li><a href="Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li>

<li><a href="Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</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="Cartesian_product" title="Cartesian product">Cartesian product</a></li>

<li><a href="Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li>

<li><a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a></li>

<li><a href="Disjoint_union" title="Disjoint union">Disjoint union</a></li>

<li><a href="List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li>

<li><a href="Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li>

<li><a href="Power_set" title="Power set">Power set</a></li>

<li><a href="Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li>

<li><a href="Union_(set_theory)" title="Union (set theory)">Union</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></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="Almost" title="Almost">Almost</a></li>

<li><a href="Cardinality" title="Cardinality">Cardinality</a></li>

<li><a href="Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="Large_cardinal" title="Large cardinal">large</a>)</li>

<li><a href="Class_(set_theory)" title="Class (set theory)">Class</a></li>

<li><a href="Constructible_universe" title="Constructible universe">Constructible universe</a></li>

<li><a href="Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li>

<li><a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li>

<li><a href="Element_(mathematics)" title="Element (mathematics)">Element</a>

<ul><li><a href="Ordered_pair" title="Ordered pair">ordered pair</a></li>

<li><a href="Tuple" title="Tuple">tuple</a></li></ul></li>

<li><a href="Family_of_sets" title="Family of sets">Family</a></li>

<li><a href="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li>

<li><a href="Bijection" title="Bijection">One-to-one correspondence</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal number</a></li>

<li><a href="Transfinite_induction" title="Transfinite induction">Transfinite induction</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="Set_(mathematics)" title="Set (mathematics)">Set</a> types</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="Amorphous_set" title="Amorphous set">Amorphous</a></li>

<li><a href="Countable_set" title="Countable set">Countable</a></li>

<li><a href="Empty_set" title="Empty set">Empty</a></li>

<li><a href="Finite_set" title="Finite set">Finite</a> (<a href="Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li>

<li><a href="Filter_(set_theory)" title="Filter (set theory)">Filter</a>

<li><a href="Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li>

<li><a href="Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li>

<li><a href="Fuzzy_set" title="Fuzzy set">Fuzzy</a></li>

<li><a href="Infinite_set" title="Infinite set">Infinite</a> (<a href="Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li>

<li><a href="Computable_set" title="Computable set">Recursive</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li>

<li><a href="Subset" title="Subset">Subset · Superset</a></li>

<li><a href="Transitive_set" title="Transitive set">Transitive</a></li>

<li><a href="Uncountable_set" title="Uncountable set">Uncountable</a></li>

<li><a href="Universal_set" title="Universal set">Universal</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</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="Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li>

<li><a href="Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li>

<li><a href="Naive_set_theory" title="Naive set theory">Naive</a></li>

<li><a href="Cantor's_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul>

<ul><li><a href="Zermelo_set_theory" title="Zermelo set theory">Zermelo</a>

<ul><li><a href="General_set_theory" title="General set theory">General</a></li></ul></li>

<li><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a>

<ul><li><a href="New_Foundations" title="New Foundations">New Foundations</a></li></ul></li>

<li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a>

<ul><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>

<ul><li><a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li>

<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li>

<li><a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></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="Russell's_paradox" title="Russell's paradox">Russell's paradox</a></li>

<li><a href="Suslin's_problem" title="Suslin's problem">Suslin's problem</a></li>

<li><a href="Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theorists</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="Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li>

<li><a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li>

<li><a href="Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li>

<li><a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li>

<li><a href="Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li>

<li><a href="Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li>

<li><a href="Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a></li>

<li><a href="Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li>

<li><a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li>

<li><a href="Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li>

<li><a href="Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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