Definition:Boolean Function
Contents |
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.
Source of Name
This entry was named for George Boole.
Sources
- Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$
