Identity function

Not to be confused with Null function or Empty 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.

Graph of the identity function on the real numbers

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 (its codomain is also its range), so it is bijective.^[2]

The identity function f on X is often denoted by id_X.

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 : X → Y is any function, then f ∘ id_X = f = id_Y ∘ f, where "∘" denotes function composition.^[4] In particular, id_X 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,^[5] 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.^[6]
• In an n-dimensional vector space the identity function is represented by the identity matrix I_n, regardless of the basis chosen for the space.^[7]
• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.^[8]
• 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 C₁).^[9]
• In a topological space, the identity function is always continuous.^[10]
• The identity function is idempotent.^[11]

See also

• Identity matrix
• Inclusion map
• Indicator function

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. Nel, Louis (2016). Continuity Theory. p. 21. doi:10.1007/978-3-319-31159-3. ISBN 978-3-319-31159-3.
5. 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
6. Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
7. T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.
8. D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.
9. James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1-85233-934-9
10. 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.
11. Conferences, University of Michigan Engineering Summer (1968). Foundations of Information Systems Engineering. we see that an identity element of a semigroup is idempotent.