Graph of the identity function on the
real numbers
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
, unchanged. That is, when is the identity function, the
equality is true for all values of to which can be applied.
Definition
Formally, if is a
set, the identity function on is defined to be a function with as its
domain and
codomain, satisfying
In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an
injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
as well as a
surjective function, so it is
bijective.
The identity function on is often denoted by .
In
set theory, where a function is defined as a particular kind of
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
, the identity function is given by the
identity relation, or ''diagonal'' of .
Algebraic properties
If is any function, then we have (where "∘" denotes
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
). In particular, is the
identity element of the
monoid of all functions from to (under function composition).
Since the identity element of a monoid is
unique, one can alternately define the identity function on to be this identity element. Such a definition generalizes to the concept of an
identity morphism in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, where the
endomorphisms of need not be functions.
Properties
*The identity function is a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
when applied to
vector spaces.
*In an -
dimensional vector space the identity function is represented by the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, 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
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
has as its
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
the
trivial group containing only this isometry (symmetry type ).
*In a
topological space, the identity function is always
continuous.
*The identity function is
idempotent.
See also
*
Identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
*
Inclusion map
References
{{DEFAULTSORT:Identity Function
Functions and mappings
Elementary mathematics
Basic concepts in set theory
Types of functions
1 (number)
