View source for Boolean-valued function

{{Short description|Function that outputs either true or false}}
{{Cleanup rewrite|date=March 2011}}
{{Functions}}

A '''Boolean-valued function''' (sometimes called a [[Predicate (logic)|predicate]] or a [[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 value]]s, for example, 0 = [[false (logic)|false]] and 1 = [[truth value|true]], i.e., a single [[bit]] of [[information]].

In the [[formal science]]s, [[mathematics]], [[mathematical logic]], [[statistics]], and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic 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 [[semantics|formal semantic]] theories of [[truth]], a '''truth predicate''' is a predicate on the [[Sentence (mathematical logic)|sentence]]s of a [[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]].

==See also==
{{colbegin|colwidth=18em}}
* [[Bit]]
* [[Boolean data type]]
* [[Boolean algebra (logic)]]
* [[Boolean domain]]
* [[Boolean logic]]
* [[Propositional calculus]]
* [[Truth table]]
* [[Logic minimization]]
* [[Indicator function]]
* [[Predicate (logic)|Predicate]]
* [[Proposition]]
* [[Boolean function]]
{{colend}}

==References==
* [[Frank Markham Brown|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.
* [[Zvi Kohavi|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.
* [[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.
* [[Marvin L. Minsky|Minsky, Marvin L.]], and [[Seymour A. Papert|Papert, Seymour, A.]] (1988), ''[[Perceptrons]], An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969.  Revised, 1972.  Expanded edition, 1988.

[[Category:Boolean algebra]]