Identity function

Graph of the identity function on the real numbers

In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In 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.^[1]

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 id_M.

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 property

If f : M → N is any function, then we have f ∘ id_M = f = id_N ∘ f (where "∘" 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.

Properties

• The identity function is a linear operator, when applied to vector spaces.^[2]
• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.^[3]
• In an n-dimensional vector space the identity function is represented by the identity matrix I_n, regardless of the basis.^[4]
• 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).^[5]

See also

• Inclusion map

References