Identity (mathematics)
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In mathematics, identity can refer to an equality that remains true regardless of the values of any variables that appear within it. Alternatively, in algebra, an identity or identity element of a set S with a binary operation is an element e which combined with any element s of S produces s. Yet a third meaning is that an identity is a function f from a set S to itself, such that f(x) = x for all x in S.
A common example of the first meaning is the trigonometric identity
- <math>( \sin \theta)^2 + ( \cos \theta)^2 = 1,\,<math>
which is true for all values of <math>\theta<math>.
A common example of the second meaning is addition in the real numbers, where 0 is the identity. This means that for all <math>a\in\Bbb{R}<math>,
- <math>0 + a = a,\,<math>
- <math>a + 0 = a,\,<math>
and
- <math>0 + 0 = 0.\,<math>
A common example of the third meaning is the identity permutation, which sends each element of the set { 1, 2, ..., n } to itself.
See also list of mathematical identities.