Mathematical Identities

Mathematical Identities

In mathematics, the term identity has several different important meanings:

• An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of the involved variables. Definitions are often indicated by the 'triple bar' symbol ≡, such as x2 ≡ x·x. The symbol ≡ can also be used with other meanings, but these can usually be interpreted in some way as a definition, or something which is otherwise tautologically true (for example, a congruence relation).

• 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.

• 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.