# Identity function

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

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.[1]

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, so it is bijective.[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.[3]

## Algebraic properties

If f : XY is any function, then we have f ∘ idX = f = idYf (where "∘" denotes function composition). 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,[4] 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.

## References

1. ^ Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978-0-8176-3248-9
2. ^ Mapa, Sadhan Kumar (7 April 2014). Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36. ISBN 978-93-80663-24-1.
3. ^ 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...
4. ^ 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
5. ^ Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
6. ^ T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.
7. ^ D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.
8. ^ James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1-85233-934-9
9. ^ 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.
10. ^ Conferences, University of Michigan Engineering Summer (1968). Foundations of Information Systems Engineering. we see that an identity element of a semigroup is idempotent.