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Milds # Identity function

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.

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## Definition

Formally, if M is a set, the identity function f on M is defined to be a function with M as its domain and codomain, satisfying

f(X) = X   for all elements X in M.

In other words, the function value f(X) in the codomain M is always the same as the input element X in the domain M. The identity function on M is clearly an injective function as well as a surjective function, so it is 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 properties

If f : MN is any function, then we have f ∘ idM = f = idNf (where "∘" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M (under function composition).

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.