s
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was used as its

argument
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...

. That is, for being identity, the equality holds for all .
Definition

Formally, if is a set, the identity function on is defined to be that function withdomain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

and codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

which satisfies
: for all elements in .
In other words, the function value in (that is, the codomain) is always the same input element of (now considered as the domain). The identity function on is clearly an injective function
In , an injective function (also known as injection, or one-to-one function) is a that maps elements to distinct elements; that is, implies . In other words, every element of the function's is the of one element of its . The term must no ...

as well as a surjective function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, so it is bijective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
The identity function on is often denoted by .
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...

, 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 elem ...

, the identity function is given by the identity relation
Identity may refer to:
Social sciences
* Identity (social science)
Identity is the qualities, beliefs, personality, looks and/or expressions that make a person (self-identity as emphasized in psychology)
or group (collective identity as p ...

, or ''diagonal'' of .
Algebraic properties

If is any function, then we have (where "∘" denotesfunction composition
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

). In particular, is the identity element
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of the monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...

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 mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

in category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

, where the endomorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s of need not be functions.
Properties

*The identity function is alinear operator
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

when applied to vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s.
*In an -dimensional
File:Dimension levels.svg, thumb
, 236px
, The first four spatial dimensions, represented in a two-dimensional picture.
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum numb ...

vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

the identity function is represented by the identity matrix
In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

, regardless of the basis
Basis may refer to:
Finance and accounting
*Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items.
Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...

chosen for the space.
*The identity function on the positive integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s is a completely multiplicative functionIn number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime n ...

(essentially multiplication by 1), considered in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

.
*In a metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

the identity function is trivially an isometry
In mathematics, an isometry (or congruence (geometry), congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be Bijection, bijective. "We shall find it convenient to use the wor ...

. An object without any symmetry
Symmetry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

has as its symmetry group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...

the trivial groupIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

containing only this isometry (symmetry type ).
*In a topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

, the identity function is always continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

.
*The identity function is idempotent
Idempotence (, ) is the property of certain operations in mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

.
See also

*Identity matrix
In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

* Inclusion map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

References

{{DEFAULTSORT:Identity Function Functions and mappings Elementary mathematicsBasic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...

Types of functions
1 (number)