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In mathematics, a function that always returns the same value that was used as its argument From Wikipedia, the free encyclopedia
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.
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]
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.
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