Identity function

File:Function-x.svg

Graph of the identity function on the real numbers

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, unchanged. That is, when $f$ is the identity function, the equality $f(x)=x$ is true for all values of $x$ to which $f$ can be applied.

Definition

Formally, if $X$ is a set, the identity function $f$ on $X$ is defined to be a function with $X$ as its domain and codomain, satisfying

$f(x)=x$ for all elements $x$ in $X$.

In other words, the function value $f(x)$ in the codomain $X$ is always the same as the input element $x$ in the domain $X$. The identity function on $X$ is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.

The identity function $f$ on $X$ is often denoted by ${\mathrm{i}\mathrm{d}}_X$.

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 $X$.

Algebraic properties

If $f:X\to Y$ is any function, then $f\circ {\mathrm{i}\mathrm{d}}_X=f={\mathrm{i}\mathrm{d}}_Y\circ f$, where "$\circ$" denotes function composition. In particular, ${\mathrm{i}\mathrm{d}}_X$ is the identity element of the monoid of all functions from $X$ to $X$ (under function composition).

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.
• In an $n$-dimensional vector space the identity function is represented by the identity matrix $I_n$, 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 containing only this isometry (symmetry type ${\mathrm{C}}_1$).
• 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