Identity: A composition in a set is said to admit of an identity if these 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 identity of multiplication for (set of integers), (set of rational numbers, (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