# 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.

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### Logical operators

Template:Col-breakTemplate:Col-breakTemplate:Col-end### Related topics

- Propositional calculus
- Sole sufficient operator
- Truth table
- Universe of discourse
- Zeroth order logic

### Relational concepts

### Information, Inquiry

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- Differential Propositional Calculus
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