# Boolean-valued function

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Boolean-valued function

A boolean-valued function, in some usages 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.

A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, for example, 0 = false and 1 = true.

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.

References

* Brown, Frank Markham (2003), "Boolean Reasoning: The Logic of Boolean Equations", 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

* Kohavi, Zvi (1978), "Switching and Finite Automata Theory", 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.

* 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.

ee also

* Boolean algebra (logic)
* Boolean domain
* Boolean logic
* Propositional calculus
* Truth table
* Logic minimization

Equivalent concepts

* Characteristic function
* Indicator function
* Predicate, in some senses.
* Proposition, in some senses.

Related concepts

* Finitary boolean function
* Boolean function

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