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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="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="Injective_function" title="Injective function">Injective</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>A <b>Boolean-valued function</b> (sometimes called a <a href="Predicate_(logic)" title="Predicate (logic)">predicate</a> or a <a href="Proposition" title="Proposition">proposition</a>) is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of the type f : X → <b>B</b>, where X is an arbitrary <a href="Set_(mathematics)" title="Set (mathematics)">set</a> and where <b>B</b> is a <a href="Boolean_domain" title="Boolean domain">Boolean domain</a>, i.e. a generic two-element set, (for example <b>B</b> = {0, 1}), whose elements are interpreted as <a href="Logical_value" class="mw-redirect" title="Logical value">logical values</a>, for example, 0 = <a href="False_(logic)" title="False (logic)">false</a> and 1 = <a href="Truth_value" title="Truth value">true</a>, i.e., a single <a href="Bit" title="Bit">bit</a> of <a href="Information" title="Information">information</a>.
</p><p>In the <a href="Formal_science" title="Formal science">formal sciences</a>, <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>, <a href="Statistics" title="Statistics">statistics</a>, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, <a href="Indicator_function" title="Indicator function">indicator function</a>, predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding <a href="Semiotic" class="mw-redirect" title="Semiotic">semiotic</a> sign or syntactic expression.
</p><p>In <a href="Semantics" title="Semantics">formal semantic</a> theories of <a href="Truth" title="Truth">truth</a>, a <b>truth predicate</b> is a predicate on the <a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentences</a> of a <a href="Formal_language" title="Formal language">formal language</a>, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final <a href="Truth_value" title="Truth value">truth value</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
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<ul><li><a href="Bit" title="Bit">Bit</a></li>
<li><a href="Boolean_data_type" title="Boolean data type">Boolean data type</a></li>
<li><a href="Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra (logic)</a></li>
<li><a href="Boolean_domain" title="Boolean domain">Boolean domain</a></li>
<li><a href="Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a></li>
<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li>
<li><a href="Truth_table" title="Truth table">Truth table</a></li>
<li><a href="Logic_minimization" class="mw-redirect" title="Logic minimization">Logic minimization</a></li>
<li><a href="Indicator_function" title="Indicator function">Indicator function</a></li>
<li><a href="Predicate_(logic)" title="Predicate (logic)">Predicate</a></li>
<li><a href="Proposition" title="Proposition">Proposition</a></li>
<li><a href="Boolean_function" title="Boolean function">Boolean function</a></li></ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li>Brown, Frank Markham (2003), <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=QPzCAgAAQBAJ">Boolean Reasoning: The Logic of Boolean Equations</a></i>, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.</li>
<li>Kohavi, Zvi (1978), <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=jZIxam8Rb9AC">Switching and Finite Automata Theory</a></i>, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978. 3rd edition, McGraw–Hill, 2010.</li>
<li><a href="Robert_R._Korfhage" title="Robert R. Korfhage">Korfhage, Robert R.</a> (1974), <i>Discrete Computational Structures</i>, Academic Press, New York, NY.</li>
<li><a href="Mathematical_Society_of_Japan" title="Mathematical Society of Japan">Mathematical Society of Japan</a>, <i>Encyclopedic Dictionary of Mathematics</i>, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM.</li>
<li><a href="Marvin_L._Minsky" class="mw-redirect" title="Marvin L. Minsky">Minsky, Marvin L.</a>, and <a href="Seymour_A._Papert" class="mw-redirect" title="Seymour A. Papert">Papert, Seymour, A.</a> (1988), <i><a href="Perceptrons" class="mw-redirect" title="Perceptrons">Perceptrons</a>, An Introduction to Computational Geometry</i>, MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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