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In 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 f is the identity function, the equality f(x) = x is true for all values of x to which f can be applied.

• Definition

• Algebraic properties

• Properties

• See also

• References

Formally, if X is a set, the identity function f on X is defined to be a function with X as its domain and codomain, satisfying

f

(

x

) =

x

for all elements

x

in

X

.

In other words, the function value f(x) in the codomain X is always the same as the input element x in the domain X. The identity function on X is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.cite-ref-2[2]

The identity function f on X is often denoted by idX.

In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of X.cite-ref-3[3]

If f : X → Y is any function, then f ∘ idX = f = idY ∘ f, where "∘" denotes function composition.cite-ref-4[4] In particular, idX is the identity element of the monoid of all functions from X to X (under function composition).

Since the identity element of a monoid is unique,cite-ref-5[5] one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.

• The identity function is a linear operator when applied to vector spaces.cite-ref-6[6]

• In an n-dimensional vector space the identity function is represented by the identity matrix In, regardless of the basis chosen for the space.cite-ref-7[7]

• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.cite-ref-8[8]

• In a metric space the identity function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group containing only this isometry (symmetry type C1).cite-ref-9[9]

• In a topological space, the identity function is always continuous.cite-ref-10[10]

• The identity function is idempotent.cite-ref-11[11]

• Identity matrix

• Inclusion map

• Indicator function

cite-note-11. citerefknapp2006Knapp, Anthony W. (2006). Basic algebra. Springer. ISBN 978-0-8176-3248-9.

cite-note-22. ↑ citerefmapa2014Mapa, Sadhan Kumar (7 April 2014). Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36. ISBN 978-93-80663-24-1.

cite-note-33. ↑ 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...

cite-note-44. ↑ citerefnel2016Nel, Louis (2016). Continuity Theory. Cham: Springer. p. 21. doi:10.1007/978-3-319-31159-3. ISBN 978-3-319-31159-3.

cite-note-55. ↑ citerefrosalesgarc-a-s-nchez1999Rosales, 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

cite-note-66. ↑ citerefanton2005Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International

cite-note-77. ↑ citereft-s-shores2007T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.

cite-note-88. ↑ citerefd-marshalle-odellm-starbird2007D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.

cite-note-99. ↑ citerefanderson2007Anderson, James W. (2007). Hyperbolic geometry. Springer undergraduate mathematics series (2. ed., corr. print ed.). London: Springer. ISBN 978-1-85233-934-0.

cite-note-1010. ↑ citerefconover2014Conover, 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.

cite-note-1111. ↑ citerefconferences1968Conferences, University of Michigan Engineering Summer (1968). Foundations of Information Systems Engineering. we see that an identity element of a semigroup is idempotent.