Identity function

An identity function is a function which maps a set S to itself. In particular, a function id maps any element (such as numbers) to themselves.

Written another way, is the function from X → X, f(x) = x, which maps every element x to itself.

In elementary math, the main example is f(x) = x with the set of numbers.

Notation

The function id_S typically denotes the identify function of a set S, the S in the subscript denotes the set S.

Algebraic properties

Given two sets X and Y with function f from X → Y, we have the two identities id_X and id_Y, as well, having "○" denote function composition. We have the following

$f\circ id_{X}=f=id_{Y}\circ f$