# Identity and Inverse

** 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.

__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 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