In mathematics, the term identity has several different important meanings:

• An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. For this, the symbol ? is sometimes used. (However, this can be ambiguous since the same symbol can also be used with different meanings, for example for a congruence relation.)
• In algebra, an identity or identity element of a set S with a binary operation · is an element e that, when combined with any element x of S, produces that same x. That is, for all x in S.
  • The identity function from a set S to itself, often denoted \mathrm{id} or \mathrm{id}_S, s the function such that \mathrm{id}(x) = x for all x in S. This function serves as the identity element in the set of all functions from S to itself with respect to function composition.
  • In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. This matrix serves as the identity with respect to matrix multiplication.

Examples

Identity relation

A common example of the first meaning is the trigonometric identity

\sin ^2 \theta + \cos ^2 \theta = 1\,

which is true for all real values of \theta (since the real numbers \Bbb{R} are the domain of sin and cos), as opposed to

\cos \theta = 1,\,

which is true only for some values of \theta, not all. For example, the latter equation is true when \theta = 0,\,, false when \theta = 2\,

See also list of mathematical identities.

Identity element

The concepts of "additive identity" and "multiplicative identity" are central to the Peano axioms. The number 0 is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all a\in\Bbb{R},

0 + a = a,\,

a + 0 = a,\, and

0 + 0 = 0.\,

Similarly, The number 1 is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all a\in\Bbb{R},

1 \times a = a,\,

a \times 1 = a,\, and

1 \times 1 = 1.\,

Identity function

A common example of an identity function is the identity permutation, which sends each element of the set \{ 1, 2, \ldots, n \} to itself.

Comparison

These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the set of permutations of \{ 1, 2, \ldots, n \} under composition.

External links

• EquationSolver - A webpage that can test a suggested identity and return a true/false "verdict".

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