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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.
- 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, e•x = x•e = 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 or , is the function which maps every element to itself. In other words, for all x in S. This function is the identity element of the composition of functions.
A common example of the first meaning is the trigonometric identity
which is true for all complex values of (since the complex numbers are the domain of sin and cos), as opposed to
which is true only for some values of , not all. For example, the latter equation is true when false when .
See also list of mathematical identities.
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,
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,
A common example of an identity function is the identity permutation, which sends each element of the set to itself or to itself in natural order.
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.