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Identity (mathematics)

For other senses of this word, see identity.

In mathematics, the term identity has several important uses:

• An identity is an equality that remains true even if you change all the variables that are used in that equality.[1][2]

An equality in mathematical sense is only true under more particular conditions. For this, the symbol ≡ is sometimes used (note, however, that the same symbol can also be used for a congruence relation as well.)

• In algebra, an identity or identity element of a set S with an operation is an element which, when combined with any element s of S, produces s itself. In a group (an algebraic structure), this is often denoted by the symbol $e$.[3]
• The identity function (or identity map) 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.[4]
• 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. It is often denoted by the symbol $I$.[3]

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 both sine and cosine), as opposed to

$\cos \theta = 1,\,$

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

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.

References

1. "The Definitive Glossary of Higher Mathematical Jargon" (in en-US). 2019-08-01.
2. Weisstein, Eric W.. "Identity Map" (in en).

Other websites

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