# Identity (mathematics)

(Redirected from Algebraic identity)

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 cos2(x) + sin2(x) = 1 are identities. Identities were sometimes indicated by the triple bar symbol ≡ instead of the equals sign =, but this is no longer a common usage.[citation needed]
• 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, ex = xe = x for all x in S. An example of this is the identity matrix when S is the set of square matrices of a particular size and the binary operation is matrix multiplication.
• 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 $\theta = 2\,$.

### Identity element

The number 0 is the additive identity (identity element for the binary operation of addition) for integers, real numbers, and complex numbers. For the real numbers, for all $a\in\Bbb{R},$

$0 + a = a,\,$
$a + 0 = a,\,$ and observe that
$0 + 0 = 0.\,$

In more abstract settings, when an additive identity exists for a binary operation on a set, the symbol 0 is often used for this element unless there is a more specialized symbol in the set.

Similarly, the number 1 is the multiplicative identity (identity element for the binary operation of multiplication) 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 observe that
$1 \times 1 = 1.\,$

### 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.