Identity function


In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. That is, for being identity, the equality holds for all.

Definition

Formally, if is a set, the identity function on is defined to be that function with domain and codomain which satisfies
In other words, the function value in is always the same input element of . The identity function on is clearly an injective function as well as a surjective function, so it is also bijective.
The identity function on is often denoted by.
In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of.

Algebraic properties

If is any function, then we have . In particular, is the identity element of the monoid of all functions from to.
Since the identity element of a monoid is unique, one can alternately define the identity function on to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of need not be functions.

Properties