Let $R$ be the relation defined in the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ by $R = \{(a, b) : \text{both } a \text{ and } b \text{ are either odd or even}\}$. Show that $R$ is an equivalence relation. Further,show that all the elements of the subset $\{1, 3, 5, 7\}$ are related to each other and all the elements of the subset $\{2, 4, 6\}$ are related to each other,but no element of the subset $\{1, 3, 5, 7\}$ is related to any element of the subset $\{2, 4, 6\}$.

(A) $1$. Reflexivity: For any $a \in A$,$a$ is either odd or even. Thus,$(a, a) \in R$. So,$R$ is reflexive.
$2$. Symmetry: If $(a, b) \in R$,then $a$ and $b$ are both odd or both even. This implies $b$ and $a$ are both odd or both even,so $(b, a) \in R$. Thus,$R$ is symmetric.
$3$. Transitivity: If $(a, b) \in R$ and $(b, c) \in R$,then $a, b$ have the same parity and $b, c$ have the same parity. Thus,$a$ and $c$ have the same parity,so $(a, c) \in R$. Thus,$R$ is transitive.
Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.
$4$. Subset analysis: All elements in $\{1, 3, 5, 7\}$ are odd,so any pair $(a, b)$ from this set satisfies the condition,meaning they are related. Similarly,all elements in $\{2, 4, 6\}$ are even,so they are related to each other. However,an odd number and an even number cannot be related,so no element from $\{1, 3, 5, 7\}$ is related to any element from $\{2, 4, 6\}$.