# Boolean-valued function

This page belongs to resource collections on Logic and Inquiry.

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 $$[k] = \{ 1, 2, \ldots k \}.$$