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.
In other words, the function value f(X) in the codomain M is always the same as the input element X in the domain M. The identity function on M is clearly an injective function as well as a surjective function, so it is bijective.
The identity function f on M is often denoted by idM.
If f : M → N is any function, then we have f ∘ idM = f = idN ∘ f (where "∘" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M (under function composition).
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.
- The identity function is a linear operator when applied to vector spaces.
- In an n-dimensional vector space the identity function is represented by the identity matrix In, 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 C1).
- In a topological space, the identity function is always continuous.
- The identity function is idempotent.
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...then the diagonal set determined by M is the identity relation...
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The element 0 is usually referred to as the identity element and if it exists, it is unique
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we see that an identity element of a semigroup is idempotent.