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Boolean-valued function

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 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.

Examples

A binary sequence is a boolean-valued function $$f : \mathbb{N}^+ \to \mathbb{B}$$, where $$\mathbb{N}^+ = \{ 1, 2, 3, \ldots \},$$. In other words, $$f\!$$ is an infinite sequence of 0's and 1's.

A binary sequence of length $$k\!$$ is a boolean-valued function $$f : [k] \to \mathbb{B}$$, where $$[k] = \{ 1, 2, \ldots k \}.$$

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.

Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.