Identity and Inverse

Identity: A composition $*$ in a set $G$ is said to admit of an identity if these 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 identity of multiplication for $\mathbb{Z}$ (set of integers), $\mathbb{Q}$ (set of rational numbers, $\mathbb{R}$ (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 operation 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.