Let $\mathrm{R}$ be the relation defined in the set $\mathrm{A}=\{1{,}2{,}3{,}4{,}5{,}6{,}7\}$ by $\mathrm{R}=\{(a, b)$ : both $a$ and $b$ are either odd or even $\}$. Show that $\mathrm{R}$ 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 $\{2{,}4{,}6\}$. Solution Given any element $a$ in A, both $a$ and $a$ must be either odd or even, so that $(a, a) \in \mathrm{R}$. Further, $(a, b) \in \mathrm{R} \Rightarrow$ both $a$ and $b$ must be either odd or even $\Rightarrow(b, a) \in \mathrm{R}$. Similarly, $(a, b) \in \mathrm{R}$ and $(b, c) \in \mathrm{R} \Rightarrow$ all elements $a, b, c$, must be either even or odd simultaneously $\Rightarrow(a, c) \in \mathrm{R}$. Hence, $\mathrm{R}$ is an equivalence relation. Further, all the elements of $\{1{,}3{,}5{,}7\}$ are related to each other, as all the elements of this subset are odd. Similarly, all the elements of the subset $\{2{,}4{,}6\}$ are related to each other, as all of them are even. Also, no element of the subset $\{1{,}3{,}5{,}7\}$ can be related to any element of $\{2{,}4{,}6\}$, as elements of $\{1{,}3{,}5{,}7\}$ are odd, while elements of $\{2{,}4{,}6\}$ are even.

Step 1: Check for Reflexivity

A relation $R$ is reflexive if every element is related to itself. For any element $a$ in set $A$, both $a$ and $a$ are either odd or even (they are the same number). Therefore, $(a, a)$ satisfies the condition for relation $R$, meaning $R$ is reflexive.

Step 2: Check for Symmetry

A relation $R$ is symmetric if whenever $(a, b)$ is in $R$, then $(b, a)$ is also in $R$. If $(a, b) \in R$, it means both $a$ and $b$ are either odd or even. This condition is symmetrical, meaning if $a$ and $b$ are both odd/even, then $b$ and $a$ are also both odd/even. Hence, $(b, a) \in R$, and $R$ is symmetric.

Step 3: Check for Transitivity

A relation $R$ is transitive if whenever $(a, b)$ is in $R$ and $(b, c)$ is in $R$, then $(a, c)$ is also in $R$. If $(a, b) \in R$, then $a$ and $b$ are both odd or both even. If $(b, c) \in R$, then $b$ and $c$ are both odd or both even. Combining these, it implies that $a$, $b$, and $c$ must all be either odd or all even. Therefore, $(a, c)$ must also satisfy the condition (both $a$ and $c$ are either odd or even), so $(a, c) \in R$. Thus, $R$ is transitive.

Step 4: Conclusion for Equivalence Relation

Since the relation $R$ satisfies all three properties: reflexivity, symmetry, and transitivity, it is an equivalence relation.

Step 5: Analyze Subsets

Consider the subset $S_1 = \{1{,}3{,}5{,}7\}$. All elements in $S_1$ are odd. According to the definition of $R$, any two elements from $S_1$ will be related to each other because they are both odd. Similarly, for the subset $S_2 = \{2{,}4{,}6\}$, all elements are even. Thus, any two elements from $S_2$ will be related to each other because they are both even.

Step 6: Analyze Relation Between Subsets

If we take an element $a$ from $S_1$ (which is odd) and an element $b$ from $S_2$ (which is even), then $a$ and $b$ are not both odd or both even. One is odd and the other is even. Therefore, $(a, b)$ does not satisfy the condition for relation $R$, meaning no element from $S_1$ is related to any element from $S_2$.

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