For other senses of this word, see identity (disambiguation).

In mathematics, the term identity has several important uses:

• identity can refer to 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. The symbol ≡ is sometimes used to indicate a mathematical identity (or a congruence relation).
• 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.
• The identity function from a set S to itself, often denoted $\backslash mathrm\{id\}$ or $\backslash mathrm\{id\}\_S$, such that $\backslash mathrm\{id\}(x)=x$ for all x in S.
• 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.

Examples

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A common example of the first meaning is the trigonometric identity

$(\; \backslash sin\; \backslash theta)^2\; +\; (\; \backslash cos\; \backslash theta)^2\; =\; 1,\backslash ,$

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

$\backslash cos\; \backslash theta\; =\; 1,\backslash ,$

which is true only for values of $\backslash theta$ in a subset of the domain.

A common example of an identity element is the number 0 among the real numbers under addition. This means that, for all $a\backslash in\backslash Bbb\{R\},$

$0\; +\; a\; =\; a,\backslash ,$

$a\; +\; 0\; =\; a,\backslash ,$ and

$0\; +\; 0\; =\; 0.\backslash ,$

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

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

See also list of mathematical identities.

Elementary algebra | Mathematical identities

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