Identity 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.

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

range), so it is bijective.^[2]

The identity function $f$ on $X$ is often denoted by $id X$ .

In

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

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

ISBN 978-0-8176-3248-9

ISBN 978-93-80663-24-1
.

ISBN 978-0-8218-1425-3
. ...then the diagonal set determined by M is the identity relation...

ISBN 978-3-319-31159-3
.

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

ISBN 978-038-733-195-9
.

ISBN 978-0883857519
.

ISBN 1-85233-934-9

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.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Identity_function&oldid=1217080195"

[1] ISBN 978-0-8176-3248-9

[2] ISBN 978-93-80663-24-1
.

[3] ISBN 978-0-8218-1425-3
. ...then the diagonal set determined by M is the identity relation...

[4] ISBN 978-3-319-31159-3
.

[5] 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] ISBN 978-038-733-195-9
.

[8] ISBN 978-0883857519
.

[9] ISBN 1-85233-934-9

[10] 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.

[1]

[2]

[3]

[4]

[6]

[7]

[8]

[9]

[10]

[11]

Definition

Algebraic properties

Properties

See also

References