Identity function

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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.

Video Identity function

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.

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.

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.

Maps Identity function

Algebraic property

If f : M -> N is any function, then we have f ? id_M = f = id_N ? 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.

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Properties

• The identity function is a linear operator, when applied to vector spaces.
• The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
• In an n-dimensional vector space the identity function is represented by the identity matrix I_n, regardless of the basis.
• 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 C₁).
• In a topological space, the identity function is always continuous.

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See also

• Inclusion map

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References

Source of article : Wikipedia