Identity function explained

thumb|Graph of the identity function on the real numbers

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

is the identity function, the equality is true for all values of to which can be applied.

Definition

Formally, if

is a set, the identity function on is defined to be a function with as its domain 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 (its codomain is also its range), so it is bijective.^[1]

The identity function

on is often denoted by .

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

.^[2]

Algebraic properties

If

is any function, then , where " " denotes function composition.^[3] In particular, is the identity element of the monoid of all functions from to (under function composition).

Since the identity element of a monoid is unique,^[4] 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 vector space the identity function is represented by the identity matrix , regardless of the basis chosen for the space.^[5]

• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.^[6]
• 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

).^[7]

• In a topological space, the identity function is always continuous.^[8]
• The identity function is idempotent.^[9]
• Every map from a set of a single element to itself is necessarily the identity map.

See also

• Identity matrix
• Inclusion map
• Indicator function

Notes and References

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