Boolean-valued function: Difference between revisions

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	A '''boolean-valued function''', in some usages a '''predicate''' or a '''proposition''', is a function of the type <math>f : X \to \mathbb{B}</math>, where <math>X</math> is an arbitrary set, where <math>\mathbb{B}</math> is a generic 2-element set, typically <math>\mathbb{B} = \left \{ 0, 1 \right \}</math>, and where the latter is frequently interpreted for logical applications as <math>\mathbb{B} = \left \{ false, true \right \}</math>.		A '''boolean-valued function''', in some usages a '''predicate''' or a '''proposition''', is a function of the type <math>f : X \to \mathbb{B}</math>, where <math>X</math> is an arbitrary set, where <math>\mathbb{B}</math> is a generic 2-element set, typically <math>\mathbb{B} = \left \{ 0, 1 \right \}</math>, and where the latter is frequently interpreted for logical applications as <math>\mathbb{B} = \left \{ false, true \right \}</math>.

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	==References==		==References==

			* ] (2003), ''Boolean Reasoning'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

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

Revision as of 21:40, 28 June 2006

A boolean-valued function, in some usages a predicate or a proposition, is a function of the type $f:X\to \mathbb {B}$ , where $X$ is an arbitrary set, where $\mathbb {B}$ is a generic 2-element set, typically $\mathbb {B} =\left\{0,1\right\}$ , and where the latter is frequently interpreted for logical applications as $\mathbb {B} =\left\{false,true\right\}$ .

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, 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 (volume).

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