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# Boolean-valued function

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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.