Identity and Inverse

Identity: A composition  * in a set G is said to admit of an identity if there exists an element e \in G such that

a * e = a = e * a{\text{ }}\forall a \in G

Moreover, the element e, if it exists, is called an identity element and the algebraic structure \left( {G, * } \right) is said to have an identity element with respect to * .

Examples:

(1) If a \in \mathbb{R} the set of real numbers, then 0 (zero) is an additive identity of\mathbb{R} because

a + 0 = a = 0 + a{\text{ }}\forall a \in \mathbb{R}

\mathbb{N} the set of natural numbers has no identity element with respect to addition because 0 \notin \mathbb{N}.

 

(2) 1 is the multiplicative identity of \mathbb{N} as

a \cdot 1 = a = 1 \cdot a{\text{ }}\forall a \in \mathbb{N}

Evidently 1 is the identity of multiplication for \mathbb{Z} (set of integers), \mathbb{Q} (set of rational numbers), and \mathbb{R} (set of real numbers).

Inverse: An element a \in G is said to have its inverse with respect to certain operation  *  if there exists b \in G such that

a * b = e = b * a

e being the identity in G with respect to a.

Such an element b, usually denoted by {a^{ - 1}}, is called the inverse of a. Thus

{a^{ - 1}} * a = e = a * {a^{ - 1}},\,\,\,\,\forall a \in G

In the set of integers the inverse of an integer a with respect to ordinary addition is  - a, and in the set of non-zero rational numbers the inverse of a with respect to multiplication is 1/a which belongs to the set.