Identity Function

Get Identity Function essential facts below. View Videos or join the Identity Function discussion. Add Identity Function to your PopFlock.com topic list for future reference or share this resource on social media.
## Definition

## Algebraic properties

## Properties

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Identity Function

In mathematics, an **identity function**, also called an **identity relation** or **identity map** or **identity transformation**, is a function that always returns the same value that was used as its argument. That is, for f being identity, the equality *f*(*x*) = *x* holds for all x.

Formally, if *M* is a set, the identity function *f* on *M* is defined to be that function with domain and codomain *M* which satisfies

*f*(*x*) =*x*for all elements*x*in*M*.^{[1]}

In other words, the function value *f*(*x*) in *M* (that is, the codomain) is always the same input element *x* of *M* (now considered as the domain). The identity function on M is clearly an injective function as well as a surjective function, so it is also bijective.^{[2]}

The identity function *f* on *M* is often denoted by id_{M}.

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]}

If *f* : *M* -> *N* is any function, then we have *f* ? id_{M} = *f* = id_{N} ? *f* (where "?" denotes function composition). In particular, id_{M} is the identity element of the monoid of all functions from *M* to *M*.

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.

- The identity function is a linear operator, when applied to vector spaces.
^{[5]} - The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
^{[6]} - In an
*n*-dimensional vector space the identity function is represented by the identity matrix I_{n}, regardless of the basis.^{[7]} - In a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the trivial group only containing this isometry (symmetry type
*C*_{1}).^{[8]} - In a topological space, the identity function is always continuous.
^{[9]} - The identity function is idempotent.
^{[10]}

**^**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...

**^**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**^**D. Marshall; E. Odell; M. Starbird (2007).*Number Theory through Inquiry*. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.**^**T. S. Shores (2007).*Applied Linear Algebra and Matrix Analysis*. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.**^**James W. Anderson,*Hyperbolic Geometry*, Springer 2005, ISBN 1-85233-934-9**^**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.**^**Conferences, University of Michigan Engineering Summer (1968).*Foundations of Information Systems Engineering*.we see that an identity element of a semigroup is idempotent.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists