Identity function

Graph of the identity function on the real numbers

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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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

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]

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.

This page was last edited on 3 April 2024, at 18:36

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