Identity function

Graph of the identity function on the real numbers

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. In equations, the function is given by $f (x) = x$ .

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

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 $M$ .

Algebraic property

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, 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.^[3]
The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.^[4]
In an $n$ -dimensional vector space the identity function is represented by the identity matrix $I n$ , regardless of the basis.^[5]
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 $C 1$ ).^[6]
In a topological space, the identity function is always continuous.

References

↑ Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978-0-8176-3248-9
↑ Mapa, Sadhan Kumar. Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36. ISBN 978-93-80663-24-1.
↑ 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 038-733-195-6.
↑ James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1-85233-934-9

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Identity function

Definition

Algebraic property

Properties

See also

References