Identity function

Last updated October 26, 2024

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.

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 $f : X \to Y$ is any function, then $f \circ id X = f = id Y \circ f$ , 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 $C 1$ ).^[9]
In a topological space, the identity function is always continuous.^[10]
The identity function is idempotent.^[11]

Related Research Articles

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References

↑ 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...
↑ Nel, Louis (2016). Continuity Theory. p. 21. doi:10.1007/978-3-319-31159-3. ISBN 978-3-319-31159-3.
↑ 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
↑ T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.
↑ D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.
↑ 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.

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[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.

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v t e Function
History List of specific functions
Types by domain and codomain	$X \to 𝔹$ $𝔹 \to X$ $𝔹ⁿ \to X$ $X \to ℤ$ $ℤ \to X$ $X \to ℝ$ $ℝ \to X$ $ℝⁿ \to X$ $X \to ℂ$ $ℂ \to X$ $ℂⁿ \to X$
Classes/properties	Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions	Restriction Composition λ Inverse
Generalizations	Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
Category