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, 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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, the identity function on is defined to be a function with as its
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
* ...
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 as well as a
surjective function, so it is
bijective.
The identity function on is often denoted by .
In
set theory
Set theory is the branch of mathematical logic that studies sets, 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 mathematics, is mostly concern ...
, where a function is defined as a particular kind of
binary relation, 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 particular, is the
identity element of the
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
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, where the
endomorphisms of need not be functions.
Properties
*The identity function is a
linear operator when applied to
vector spaces.
*In an -
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
vector space the identity function is represented by the
identity matrix , regardless of the
basis chosen for the space.
*The identity function on the positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s is a
completely multiplicative function (essentially multiplication by 1), considered in
number theory.
*In a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
the identity function is trivially an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
. 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 definiti ...
has as its
symmetry group the
trivial group containing only this isometry (symmetry type ).
*In a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, 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 g ...
.
*The identity function is
idempotent.
See also
*
Identity matrix
*
Inclusion map
References
{{DEFAULTSORT:Identity Function
Functions and mappings
Elementary mathematics
Basic concepts in set theory
Types of functions
1 (number)
