# Boolean-valued function

Jump to navigation Jump to search

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