Identity: A composition in a set is said to admit of an identity if there exists an element such that
Moreover, the element , if it exists, is called an identity element and the algebraic structure is said to have an identity element with respect to.
(1) If the set of real numbers, then (zero) is an additive identity of because
the set of natural numbers has no identity element with respect to addition because .
(2) is the multiplicative identity of as
Evidently is the identity of multiplication for (set of integers), (set of rational numbers), and (set of real numbers).
Inverse: An element is said to have its inverse with respect to certain operation if there exists such that
being the identity in with respect to .
Such an element , usually denoted by , is called the inverse of . Thus