Identity function

In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. That is, for $f$ being identity, the equality $f (x) = x$ holds for all $x$ .

Graph of the identity function on the real numbers

Definition

Formally, if $M$ is a set, the identity function $f$ on $M$ is defined to be that function with domain and codomain $M$ which satisfies

f (x) = x

for all elements

x

in

M

.[1]

In other words, the function value $f (x)$ in $M$ (that is, the codomain) is always the same input element $x$ of $M$ (now considered as the domain). The identity function on $M$ is clearly an injective function as well as a surjective function, so it is also bijective.[2]

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

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 $M$ .[3]

Algebraic properties

If $f : M \to N$ is any function, then we have $f \circ id M = f = id N \circ f$ (where "∘" denotes function composition). In particular, $id M$ is the identity element of the monoid of all functions from $M$ to $M$ .

Since the identity element of a monoid is unique,[4] 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.[5]
The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.[6]
In an $n$ -dimensional vector space the identity function is represented by the identity matrix $I n$ , regardless of the basis.[7]
In a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the trivial group only containing this isometry (symmetry type $C 1$ ).[8]
In a topological space, the identity function is always continuous.[9]
The identity function is idempotent.[10]

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...
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
D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.
T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.
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] 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] D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.

[7] T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.

[8] James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1-85233-934-9

[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.

Identity function

Definition

Algebraic properties

Properties

See also

References