Identity function
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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 {\displaystyle f} is the identity function, the equality f ( x ) = x {\displaystyle f(x)=x} is true for all values of x {\displaystyle x} to which f {\displaystyle f} can be applied.
Definition
Formally, if X {\displaystyle X} is a set, the identity function f {\displaystyle f} on X {\displaystyle X} is defined to be a function with X {\displaystyle X} as its domain and codomain, satisfying
In other words, the function value f ( x ) {\displaystyle f(x)} in the codomain X {\displaystyle X} is always the same as the input element x {\displaystyle x} in the domain X {\displaystyle X}. The identity function on X {\displaystyle X} is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.
The identity function f {\displaystyle f} on X {\displaystyle X} is often denoted by i d X {\displaystyle \mathrm {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 {\displaystyle X}.
Algebraic properties
If f : X → Y {\displaystyle f:X\rightarrow Y} is any function, then f ∘ i d X = f = i d Y ∘ f {\displaystyle f\circ \mathrm {id} _{X}=f=\mathrm {id} _{Y}\circ f}, where "∘ {\displaystyle \circ }" denotes function composition. In particular, i d X {\displaystyle \mathrm {id} _{X}} is the identity element of the monoid of all functions from X {\displaystyle X} to X {\displaystyle X} (under function composition).
Since the identity element of a monoid is unique, one can alternately define the identity function on M {\displaystyle 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 {\displaystyle M} need not be functions.
Properties
- The identity function is a linear operator when applied to vector spaces.
- In an n {\displaystyle n}-dimensional vector space the identity function is represented by the identity matrix I n {\displaystyle I_{n}}, 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 C 1 {\displaystyle \mathrm {C} _{1}}).
- In a topological space, the identity function is always continuous.
- The identity function is idempotent.
- Every map from a set of a single element to itself is necessarily the identity map.