# Boolean-valued function

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Boolean-valued_function
> Markdown URL: https://mediated.wiki/source/Boolean-valued_function.md
> Source: https://en.wikipedia.org/wiki/Boolean-valued_function
> Source revision: 1351865012
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Function that outputs either true or false}}
{{mi|
{{more footnotes|date=August 2025}}
{{too technical|date=August 2025}}}}
{{Functions}}

A '''Boolean-valued function''' (sometimes called a [predicate](/source/Predicate_(logic)) or a [proposition](/source/proposition)) is a [function](/source/function_(mathematics)) of the type f : X → '''B''', where X is an arbitrary [set](/source/Set_(mathematics)) and where '''B''' is a [Boolean domain](/source/Boolean_domain), i.e. a generic two-element set, (for example '''B''' = {0, 1}), whose elements are interpreted as [logical value](/source/logical_value)s, for example, 0 = [false](/source/false_(logic)) and 1 = [true](/source/truth_value), i.e., a single [bit](/source/bit) of [information](/source/information).

In the [formal science](/source/formal_science)s, [mathematics](/source/mathematics), [mathematical logic](/source/mathematical_logic), [statistics](/source/statistics), and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, [indicator function](/source/indicator_function), predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding [semiotic](/source/semiotic) sign or syntactic expression.

In [formal semantic](/source/semantics) theories of [truth](/source/truth), a '''truth predicate''' is a predicate on the [sentence](/source/Sentence_(mathematical_logic))s of a [formal language](/source/formal_language), 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 [truth value](/source/truth_value).

==See also==
{{colbegin|colwidth=18em}}
* [Bit](/source/Bit)
* [Boolean data type](/source/Boolean_data_type)
* [Boolean algebra (logic)](/source/Boolean_algebra_(logic))
* [Boolean domain](/source/Boolean_domain)
* [Boolean logic](/source/Boolean_logic)
* [Propositional calculus](/source/Propositional_calculus)
* [Truth table](/source/Truth_table)
* [Logic minimization](/source/Logic_minimization)
* [Indicator function](/source/Indicator_function)
* [Predicate](/source/Predicate_(logic))
* [Proposition](/source/Proposition)
* [Boolean function](/source/Boolean_function)
{{colend}}

==References==
* [Brown, Frank Markham](/source/Frank_Markham_Brown) (2003), ''[https://books.google.com/books?id=QPzCAgAAQBAJ Boolean Reasoning:  The Logic of Boolean Equations]'', 1st edition, Kluwer Academic Publishers, Norwell, MA.  2nd edition, Dover Publications, Mineola, NY, 2003.
* [Kohavi, Zvi](/source/Zvi_Kohavi) (1978), ''[https://books.google.com/books?id=jZIxam8Rb9AC Switching and Finite Automata Theory]'', 1st edition, McGraw–Hill, 1970.  2nd edition, McGraw–Hill, 1978. 3rd edition, McGraw–Hill, 2010.
* [Korfhage, Robert R.](/source/Robert_R._Korfhage) (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
* [Mathematical Society of Japan](/source/Mathematical_Society_of_Japan), ''[Encyclopedic Dictionary of Mathematics](/source/Encyclopedic_Dictionary_of_Mathematics)'', 2nd edition, 2 vols., [Kiyosi Itô](/source/Kiyosi_It%C3%B4) (ed.), MIT Press, Cambridge, MA, 1993.  Cited as EDM.
* [Minsky, Marvin L.](/source/Marvin_L._Minsky), and [Papert, Seymour, A.](/source/Seymour_A._Papert) (1988), ''[Perceptrons](/source/Perceptrons), An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969.  Revised, 1972.  Expanded edition, 1988.

Category:Boolean algebra

---
Adapted from the Wikipedia article [Boolean-valued function](https://en.wikipedia.org/wiki/Boolean-valued_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Boolean-valued_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.