Identity function
From Exampleproblems
In mathematics, an identity function, also called identity map or identity transformation, is a function which does not have any effect: it always returns the same value that was used as its argument. In other words, the identity function is the function f(x) = x.
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.
The identity function f on M is often denoted by idM or 1M.
On each set M, there exists only one an identity function (so we talk of "the" identity function of M). This can be deduced from the fact that all of automorphisms on some set forms a group, which has an unique identity element.
If f : M → N is any function, then we have f o idM = f = idN o f (where "o" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M.
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 In, 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).
