Identity function

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

[1]

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]

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.

• The identity function is a linear operator when applied to vector spaces.^[5]
• In an -dimensional vector space the identity function is represented by the identity matrix , 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 ).^[8]
• In a topological space, the identity function is always continuous.^[9]
• The identity function is idempotent.^[10]
• Every map from a set of a single element to itself is necessarily the identity map.

• Identity matrix
• Inclusion map
• Indicator function

1. ^ Mapa, Sadhan Kumar (7 April 2014). Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36. ISBN 978-93-80663-24-1.
2. ^ 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...
3. ^ Nel, Louis (2016). Continuity Theory. Cham: Springer. p. 21. doi:10.1007/978-3-319-31159-3. ISBN 978-3-319-31159-3.
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. ^ Anderson, James W. (2007). Hyperbolic geometry. Springer undergraduate mathematics series (2. ed., corr. print ed.). London: Springer. ISBN 978-1-85233-934-0.
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.