Identity (mathematics)

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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 A and B produce the same value as each other regardless of what values (usually numbers) are substituted for the variables. 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 binary operation • is an element e that, when combined with any element x of the set on which the operation is defined, produces that same x. That is, ex = xe = x for all x. Examples of this are 0 for the addition, 1 for the multiplication of numbers, and also the identity matrix for the multiplication of square matrices of a fixed size.
  • 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 is the identity element of the composition of functions.


Identity relation[edit]

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[edit]

The number 0 is the additive identity (identity element for the binary operation of addition) for integers, real numbers, and complex numbers. For every number a, including 0 itself,

0 + a = a+0=a\,.

In a more general context, when a binary operation is denoted with + and has an identity, this identity is commonly denoted by the symbol 0 (zero) and called an additive identity.

Similarly, the number 1 is the identity of the multiplication of numbers. It is often called the multiplicative identity for distinguishing it from the additive identity, zero. For every number a, including 1 itself,

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

Identity function[edit]

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.


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.

See also[edit]

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