Английская Википедия:Identity function

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Файл:Function-x.svg

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 Шаблон:Mvar is the identity function, the equality Шаблон:Math is true for all values of Шаблон:Mvar to which Шаблон:Mvar can be applied.

Definition

Formally, if Шаблон:Math is a set, the identity function Шаблон:Math on Шаблон:Math is defined to be a function with Шаблон:Math as its domain and codomain, satisfying Шаблон:Bi

In other words, the function value Шаблон:Math in the codomain Шаблон:Math is always the same as the input element Шаблон:Math in the domain Шаблон:Math. The identity function on Шаблон:Mvar is clearly an injective function as well as a surjective function, so it is bijective.

The identity function Шаблон:Math on Шаблон:Math is often denoted by Шаблон:Math.

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 Шаблон:Math.

Algebraic properties

If Шаблон:Math is any function, then we have Шаблон:Math (where "∘" denotes function composition). In particular, Шаблон:Math is the identity element of the monoid of all functions from Шаблон:Math to Шаблон:Math (under function composition).

Since the identity element of a monoid is unique, one can alternately define the identity function on Шаблон:Math to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of Шаблон:Math need not be functions.

Properties

• The identity function is a linear operator when applied to vector spaces.
• In an Шаблон:Mvar-dimensional vector space the identity function is represented by the identity matrix Шаблон:Math, regardless of the basis chosen for the space.
• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
• 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 Шаблон:Math).
• In a topological space, the identity function is always continuous.
• The identity function is idempotent.

See also

• Identity matrix
• Inclusion map
• Indicator function