- Identity function
In

mathematics , an**identity function**, also called**identity map**or**identity transformation**, is a function that always returns the same value that was used as its argument. In terms ofequation s, the function is given by "f"("x") = "x".**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 others words, the function assigns to each element "x" of "M" the element "x" of "M".

The identity function "f" on "M" is often denoted by id

_{"M"}or 1_{"M"}.In terms of

set theory , where a function is defined as a particular kind ofbinary relation , the identity function is given by theidentity relation , or "diagonal" of "M".**Algebraic property**If "f" : "M" → "N" is any function, then we have "f" o id

_{"M"}= "f" = id_{"N"}o "f" (where "o" denotesfunction composition ). In particular, id_{"M"}is theidentity element of themonoid 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 anidentity morphism incategory theory , where theendomorphism s of "M" need not be functions.**Examples***The identity function is a

linear operator , when applied tovector spaces .

*The identity function on the positiveinteger s is acompletely multiplicative function (essentially multiplication by 1), considered innumber theory .

*In an "n"-dimensionalvector space the identity function is represented by theidentity matrix "I"_{"n"}, regardless of the basis.

*In ametric space the identity is trivially anisometry . An object without anysymmetry has assymmetry group the trivial group only containing this isometry (symmetry type "C_{1}).**See also***

Inclusion map

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