To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Identity function

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

• 1/3
Views:
57 278
18 893
30 794
• Identity Function & It's Graph - Learn Relations and Function
• (Abstract Algebra 1) The Identity Function
• Identity Function

## Definition

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

f(X) = X   for all elements X in M.[1]

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

The identity function f on M is often denoted by idM.

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 M.[3]

## Algebraic properties

If f : MN is any function, then we have f ∘ idM = f = idNf (where "∘" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M (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.