Question

The relation $R$ in the set $Z$ of integers given by $R=\{(a,b):2$ divides $a-b\}$.
The relation defined above is

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

Answer 1

Answer: (D)

$R$ is reflexive, as 2 divides $(a-a)$ for all $a\in Z$. Further, if $(a,b)\in R$,then 2 divides $a-b$. Therefore, 2 divides $b-a$. Hence, $(b,a)\in R$, which shows that $R$ is symmetric. Similarly, if $(a,b)\in R$ and $(b,c)\in R$, then $a-b$ and $b-c$ are divisible by 2. Now $a-c=(a-b)+(b-c)$ is even. So, $(a-c)$ is divisible by 2 . This shows that $R$ is transitive. Thus, $R$ is an equivalence relation in $Z$.