Difference between revisions of "Boolean-valued function"
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| − | A '''boolean-valued function''' is a [[function (mathematics)|function]] of the type | + | A '''boolean-valued function''' is a [[function (mathematics)|function]] of the type <math>f : X \to \mathbb{B},</math> where <math>X\!</math> is an arbitrary [[set]] and where <math>\mathbb{B}</math> is a [[boolean domain]]. |
| − | In the [[formal science]]s | + | 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]]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. | In [[semantics|formal semantic]] theories of [[truth]], a '''truth predicate''' is a predicate on the [[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. | ||
Revision as of 00:10, 22 October 2008
A boolean-valued function is a function of the type \(f : X \to \mathbb{B},\) where \(X\!\) is an arbitrary set and where \(\mathbb{B}\) is a boolean domain.
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.
See also
Equivalent concepts
- Characteristic function
- Indicator function
- Predicate, in some senses.
- Proposition, in some senses.
Related concepts
Aficionados
- See Talk:Boolean-valued function for discussions/comments regarding this article.
- See Boolean-valued function/Aficionados for those who have listed Boolean-valued function as an interest.
- See Talk:Boolean-valued function/Aficionados for discussions regarding this interest.
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