Definition:Boolean Function

Definition
A (finitary) boolean function is a function of the form $$f : \mathbb B^k \to \mathbb B$$, where:
 * $$\mathbb B = \left\{{0, 1}\right\}$$ is a boolean domain;
 * $$k\!$$ is a nonnegative integer.

In the case where $$k = 0\!$$, then $$f$$ is the constant function, and its value is simply a constant element of $$\mathbb B$$.

The boolean domain most often seen in the field of logic is $$\mathbb B = \left\{{T, F}\right\}$$, where $$T$$ stands for true and $$F$$ for false.

From Count of Boolean Functions, there are $$2^{\left({2^k}\right)}$$ boolean functions on $$k\!$$ variables.

Truth Function
A boolean function is often referred to as a truth function.

In treatments which come from the direction of symbolic logic, the following specific functional notation is sometimes used:


 * The Logical Not connective gives rise to the truth function $$f^\neg: \mathbb B \to \mathbb B$$.
 * The conjunction connective gives rise to the truth function $$f^\and: \mathbb B^2 \to \mathbb B$$.
 * The disjunction connective gives rise to the truth function $$f^\or: \mathbb B^2 \to \mathbb B$$.

And so on.

Linguistic Note
The word boolean has entered the field of computer science as a noun meaning "a variable which can take one of two values".

Note that although the modern usage renders it without a capital B, you will find that older texts use Boolean.