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Graph of the identity function on the real number
Graph</a> of the identity function on the real number
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 with
domain 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 ...

codomain
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 ...

injective function
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 ...

surjective function
, 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 ...

bijective
. 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 "∘" denotes
function 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 a
linear 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 ...

linear operator
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 ...

identity matrix
, 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 ...

number theory
. *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 ...

symmetry
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 ...

Identity matrix
*
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 mathematics
Basic 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)