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.
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(Abstract Algebra 1) The Identity Function

Identity Function
Transcription
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
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 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]}
Algebraic properties
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 (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.
Properties
 The identity function is a linear operator when applied to vector spaces.^{[5]}
 In an ndimensional vector space the identity function is represented by the identity matrix I_{n}, regardless of the basis chosen for the space.^{[6]}
 The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.^{[7]}
 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 C_{1}).^{[8]}
 In a topological space, the identity function is always continuous.^{[9]}
 The identity function is idempotent.^{[10]}
See also
References
 ^ Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 9780817632489
 ^ Mapa, Sadhan Kumar (7 April 2014). Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36. ISBN 9789380663241.
 ^ Proceedings of Symposia in Pure Mathematics. American Mathematical Society. 1974. p. 92. ISBN 9780821814253.
...then the diagonal set determined by M is the identity relation...
 ^ Rosales, J. C.; GarcíaSánchez, P. A. (1999). Finitely Generated Commutative Monoids. Nova Publishers. p. 1. ISBN 9781560726708.
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
 ^ T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 9780387331959.
 ^ D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 9780883857519.
 ^ James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1852339349
 ^ Conover, Robert A. (20140521). A First Course in Topology: An Introduction to Mathematical Thinking. Courier Corporation. p. 65. ISBN 9780486780016.
 ^ Conferences, University of Michigan Engineering Summer (1968). Foundations of Information Systems Engineering.
we see that an identity element of a semigroup is idempotent.