Identity (mathematics)

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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, e•x = x•e = 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 ${\mathbb {C}}$ 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 $a\in {\mathbb {R}},$

$0+a=a,\,$

$a+0=a,\,$ 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 $a\in {\mathbb {R}},$

$1\times a=a,\,$

$a\times 1=a,\,$ 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.

External links

A Collection of Algebraic Identities

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