View source for Boolean-valued function

{{Short description|Function that outputs either true or false}}
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A '''Boolean-valued function''' (sometimes called a predicate or a [[Proposition|proposition]]) is a [[Function (mathematics)|function]] of the type f : X → '''B''', where X is an arbitrary [[Set (mathematics)|set]] and where '''B''' is a [[Boolean domain]], i.e. a generic two-element set, (for example '''B''' = {0, 1}), whose elements are interpreted as logical values, for example, 0 = [[Philosophy:False (logic)|false]] and 1 = [[Philosophy:Truth value|true]], i.e., a single [[Bit|bit]] of [[Information|information]].

In the [[Formal science|formal science]]s, [[Mathematics|mathematics]], [[Mathematical logic|mathematical logic]], [[Statistics|statistics]], and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, [[Indicator function|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 sign or syntactic expression.

In [[Philosophy:Semantics|formal semantic]] theories of [[Truth|truth]], a '''truth predicate''' is a predicate on the [[Philosophy:Sentence (mathematical logic)|sentence]]s of a [[Formal language|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 [[Philosophy:Truth value|truth value]].

==See also==
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* [[Bit]]
* [[Boolean data type]]
* Boolean algebra (logic)
* [[Boolean domain]]
* Boolean logic
* [[Propositional calculus]]
* [[Truth table]]
* Logic minimization
* [[Indicator function]]
* Predicate
* [[Proposition]]
* [[Boolean function]]
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==References==
* Brown, Frank Markham (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 (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.
* [[Biography:Robert R. Korfhage|Korfhage, Robert R.]] (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
* Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993.  Cited as EDM.
* Minsky, Marvin L., and Papert, Seymour, A. (1988), ''Perceptrons, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969.  Revised, 1972.  Expanded edition, 1988.

[[Category:Boolean algebra]]

{{Sourceattribution|Boolean-valued function}}