 | ||
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. In equations, the function is given by f(x) = x.
Contents
Definition
Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies
f(x) = x for all elements x in M.In other words, the function value f(x) in M (that is, the codomain) is always the same input element x of M (now considered as the domain). The identity function on M is clearly an injective function as well as a surjective function, so it is also bijective.
The identity function f on M is often denoted by idM.
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 M.
Algebraic property
If f : M → N is any function, then we have f ∘ idM = f = idN ∘ f (where "∘" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M.
Since the identity element of a monoid is unique, 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.