Question

The relation $R$ defined in the set $A=\{1,2,3,4,5$, $6,7\}$ by $R=\{(a,b)$ : both $a$ and $b$ are either odd or even $\}$. Then, $R$ is

• A. symmetric
• B. transitive
• C. an equivalence relation
• D. reflexive

Answer 1

Answer: (C)

Given, any element a in $A$, both a and a must be either odd or even, so that $(a,a)\in R$. Further, $(a,b)\in R\Rightarrow$ both a and $b$ must be either odd or even $\Rightarrow (b,a)\in R$. Similarly, $(a,b)$ $\in R$ and $(b,c)\in R\Rightarrow$ all elements $a,b,c$ must be either even or odd simultaneously $\Rightarrow (a,c)\in R$. Hence, $R$ is an equivalence relation.