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 is the identity function, the equality can be applied.

Definition

is a set as its domain and codomain, satisfying

is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.

.

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.

Algebraic properties

" denotes function composition. is the identity element of the monoid (under function composition).

Since the identity element of a monoid is unique, to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms need not be functions.

Properties

• The identity function is a linear operator when applied to vector spaces.

• -dimensional vector space the identity function is represented by the identity matrix, regardless of the basis chosen for the space.

• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.

• In a metric space the identity function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group).

• In a topological space, the identity function is always continuous.

• The identity function is idempotent.

• Every map from a set of a single element to itself is necessarily the identity map.

See also

• Identity matrix

• Inclusion map

• Indicator function

References