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.

Examples:

(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  for.

In the set of integers the inverse of an integer with respect to ordinary addition operation is and in the set of non-zero rational numbers, the inverse of with respect to multiplication is  which belongs to the set.