Identity (mathematics)

For other uses, see Identity (disambiguation).

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

• An identity is an equality relation A = B, such that A and B contain some variables and give the same result when the variables are substituted by any values (usually numbers). In other words, A = B is an identity if A and B define the same functions. This means that an identity is an equality between functions that are differently defined. For example (x + y)²  =  x² + 2xy + y² and cos(x)² + sin(x)² = 1 are identities. Identities were sometimes indicated by the triple bar symbol ≡ instead of the equals sign =, but it is no more the common usage.

• 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, e·x = x·e = x for all x in S. An example of this is the identity matrix.

• The identity function from a set S to itself, often denoted $\mathrm{id}$ or $\mathrm{id}_S$, is the function which maps every element to itself. In other words, $\mathrm{id}(x) = x$ 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.

Examples

Identity relation

A common example of the first meaning is the trigonometric identity

$\sin ^2 \theta + \cos ^2 \theta \equiv 1\,$

which is true for all complex values of $\theta$ (since the complex numbers $\Bbb{C}$ 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 or $\{a_1,a_2, \ldots, a_n \}$ to itself in natural order.

Comparison

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

Also, some care is sometimes needed to avoid ambiguities: 0 is the identity element for the addition of numbers and x + 0 = x is an identity. On the other hand, the identity function f(x) = x is not the identity element for the addition or the multiplication of functions (these are the constant functions 0 and 1), but is the identity element for the function composition.

External links

• A Collection of Algebraic Identities