{{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 or a proposition) is a function of the type f : X → '''B''', where X is an arbitrary 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 = false and 1 = true, i.e., a single bit of information.
In the formal sciences, 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 formal semantic theories of truth, a '''truth predicate''' is a predicate on the sentences 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 * Proposition * Boolean function {{colend}}
==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. * 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