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EspañolIDENTITY FUNCTION
In mathematics, an 'identity function', also called 'identity map' or 'identity transformation', is a function that 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 id''M'' or 1''M''.
If ''f'' : ''M'' → ''N'' is any function, then we have ''f'' o id''M'' = ''f'' = id''N'' o ''f'' (where "o" 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.
★ 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).
★ Inclusion map
| Contents |
| Definition |
| Algebraic property |
| Examples |
| See also |
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''.
The identity function ''f'' on ''M'' is often denoted by id''M'' or 1''M''.
Algebraic property
If ''f'' : ''M'' → ''N'' is any function, then we have ''f'' o id''M'' = ''f'' = id''N'' o ''f'' (where "o" 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.
Examples
★ 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).
See also
★ Inclusion map
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