An example of a binary operation on a set of functions

Let and let F be the set of all functions from X to itself. There are four such functions and so . How each function maps elements of X is tabulated in the following table.

On F define a binary operation to be function composition. That is if u and v are functions then uv is the function defined by uv(x) = u(v(x)) for each . We shall refer to this operation as multiplication on F. The multiplication table is as follows:

This is an associative operation since function composition is associative. However, it is not commutative. The element g is a left and right identity. Note h and g are the only elements that have inverses and h is its own inverse as is g.