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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 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 or , is the function which maps every element to itself. In other words, 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.
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 the real numbers, for all
- and observe that
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
- and observe that
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.