In mathematics, an identity function, also called identity map or identity transformation, is a function which does not have any effect: it always returns the same value that was used as its argument. In other words, the identity function is the function f(x) = x.

Definition

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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.

The identity function f on M is often denoted by id_M or 1_M.

Algebraic property

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If f : M → N is any function, then we have f o id_M = f = id_N o f (where "o" denotes function composition). In particular, id_M 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.

Examples

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• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
• In an n-dimensional vector space the identity function is represented by the identity matrix I_n, regardless of the basis.
• In a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the trivial group only containing this isometry (symmetry type ''C₁).

See also

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• Inclusion map

Elementary mathematics

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