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$$ * $$.


(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.