In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f(x) = x is true for all values of x to which f can be applied.
Definition
Formally, if X is a set, the identity function f on X is defined to be a function with X as its domain and codomain, satisfying
In other words, the function value f(x) in the codomain X is always the same as the input element x in the domain X. The identity function on X is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.[2]
The identity function f on X is often denoted by idX.
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 X.[3]
Algebraic properties
If f : X → Y is any function, then f ∘ idX = f = idY ∘ f, where "∘" denotes function composition.[4] In particular, idX is the identity element of the monoid of all functions from X to X (under function composition).
Since the identity element of a monoid is unique,[5] one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.
Properties
- The identity function is a linear operator when applied to vector spaces.[6]
- In an n-dimensional vector space the identity function is represented by the identity matrix In, regardless of the basis chosen for the space.[7]
- The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.[8]
- In a metric space the identity function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group containing only this isometry (symmetry type C1).[9]
- In a topological space, the identity function is always continuous.[10]
- The identity function is idempotent.[11]