Identity (mathematics)

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 $\mathrm {id}$ or $\mathrm {id} _{S}$, such that $\mathrm {id} (x)=x$ 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

A common example of the first meaning is the trigonometric identity

$(\sin \theta )^{2}+(\cos \theta )^{2}=1,\,$

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

$\cos \theta =1,\,$

which is true only for values of $\theta$ 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 $a\in {\mathbb {R}}$,

$0+a=a\,$,

$a+0=a\,$, and

$0+0=0\,$.

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

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.

See also list of mathematical identities.