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 $f^\land: \mathbb B^2 \to \mathbb B$.
 * The disjunction connective gives rise to the truth function $f^\lor: \mathbb B^2 \to \mathbb B$.

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 (exactly) two values.

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