Identity function
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In mathematics, an identity function, also called identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In terms of equations, 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 other 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 idM.
In terms of 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 o idM = f = idN o f (where "o" 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.
Properties
- The identity function is a linear operator, when applied to vector spaces.
- 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 In, 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).