Movatterモバイル変換

[0]ホーム

Jump to content

Identity function

Edit links

From Wikipedia, the free encyclopedia

Function that returns its argument unchanged

Not to be confused withNull function orEmpty function.

Graph of the identity function on thereal numbers

Inmathematics, anidentity function, also called anidentity relation,identity map oridentity transformation, is afunction that always returns the value that was used as itsargument, unchanged. That is, when $f {\displaystyle f}$ is the identity function, theequality $f(x)=x$ is true for all values of $x {\displaystyle x}$ to which $f {\displaystyle f}$ can be applied.

Definition

[edit]

Formally, if $X {\displaystyle X}$ is aset, the identity function $f {\displaystyle f}$ on $X {\displaystyle X}$ is defined to be a function with $X {\displaystyle X}$ as itsdomain andcodomain, satisfying

f(x)=x

for all elements

x {\displaystyle x}

X {\displaystyle X}

.^[1]

In other words, the function value $f(x)$ in the codomain $X {\displaystyle X}$ is always the same as the input element $x {\displaystyle x}$ in the domain $X {\displaystyle X}$ . The identity function on $X {\displaystyle X}$ is clearly aninjective function as well as asurjective function (its codomain is also itsrange), so it isbijective.^[2]

The identity function $f {\displaystyle f}$ on $X {\displaystyle X}$ is often denoted by $\mathrm {id} _{X}$ .

Inset theory, where a function is defined as a particular kind ofbinary relation, the identity function is given by theidentity relation, ordiagonal of $X {\displaystyle X}$ .^[3]

Algebraic properties

[edit]

If $f:X\rightarrow Y$ is any function, then $f\circ \mathrm {id} _{X}=f=\mathrm {id} _{Y}\circ f$ , where " $\circ$ " denotesfunction composition.^[4] In particular, $\mathrm {id} _{X}$ is theidentity element of themonoid of all functions from $X {\displaystyle X}$ to $X {\displaystyle X}$ (under function composition).

Since the identity element of a monoid isunique,^[5] one can alternately define the identity function on $M {\displaystyle M}$ to be this identity element. Such a definition generalizes to the concept of anidentity morphism incategory theory, where theendomorphisms of $M {\displaystyle M}$ need not be functions.

Properties

[edit]

The identity function is alinear operator when applied tovector spaces.^[6]
In an $n {\displaystyle n}$ -dimensional vector space the identity function is represented by theidentity matrix $I_{n}$ , regardless of thebasis chosen for the space.^[7]
The identity function on the positiveintegers is acompletely multiplicative function (essentially multiplication by 1), considered innumber theory.^[8]
In ametric space the identity function is trivially anisometry. An object without anysymmetry has as itssymmetry group thetrivial group containing only this isometry (symmetry type $\mathrm {C} _{1}$ ).^[9]
In atopological space, the identity function is alwayscontinuous.^[10]
The identity function isidempotent.^[11]

References

[edit]

^Knapp, Anthony W. (2006).Basic algebra. Springer.ISBN 978-0-8176-3248-9.
^Mapa, Sadhan Kumar (7 April 2014).Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36.ISBN 978-93-80663-24-1.
^Proceedings of Symposia in Pure Mathematics. American Mathematical Society. 1974. p. 92.ISBN 978-0-8218-1425-3....then the diagonal set determined by M is the identity relation...
^Nel, Louis (2016).Continuity Theory. Cham: Springer. p. 21.doi:10.1007/978-3-319-31159-3.ISBN 978-3-319-31159-3.
^Rosales, J. C.; García-Sánchez, P. A. (1999).Finitely Generated Commutative Monoids. Nova Publishers. p. 1.ISBN 978-1-56072-670-8.The element 0 is usually referred to as the identity element and if it exists, it is unique
^Anton, Howard (2005),Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
^T. S. Shores (2007).Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer.ISBN 978-038-733-195-9.
^D. Marshall; E. Odell; M. Starbird (2007).Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer.ISBN 978-0883857519.
^Anderson, James W. (2007).Hyperbolic geometry. Springer undergraduate mathematics series (2. ed., corr. print ed.). London: Springer.ISBN 978-1-85233-934-0.
^Conover, Robert A. (2014-05-21).A First Course in Topology: An Introduction to Mathematical Thinking. Courier Corporation. p. 65.ISBN 978-0-486-78001-6.
^Conferences, University of Michigan Engineering Summer (1968).Foundations of Information Systems Engineering.we see that an identity element of a semigroup is idempotent.

v t e Function
History List of specific functions
Types by domain and codomain	X → 𝔹 𝔹 → X 𝔹ⁿ → 𝔹 X → ℤ ℤ → X X → ℝ ℝ → X ℝⁿ → X X → ℂ ℂ → X ℂⁿ → X
Classes/properties	Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions	Restriction Composition λ Inverse
Generalizations	Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
Category

Retrieved from "https://en.wikipedia.org/w/index.php?title=Identity_function&oldid=1335411051"

Categories:

Hidden categories:

[8]ページ先頭

Movatterモバイル変換

Definition

Algebraic properties

Properties

See also

References