Question

Let $R$ be the relation in the set $Z$ of all integers defined by $R=\{(x,y):x-y$ is an integer $\}$. Then $R$ is

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

Answer 1

Answer: (D)

Here, $R=\{(x,y):x-y$ is an integer $\}$ is a relation in the set of integers.
For reflexivity, put $y=x,x-x=0$ which is an integer for all $x\in Z$. So, $R$ is reflexive in $Z$.
For symmetry, let $(x,y)\in R$, then $(x-y)$ is an integer $\lambda$ (say) and also $y-x=-\lambda$. $(\because \lambda \in Z\Rightarrow -\lambda \in Z)$ $\therefore y-x$ is an integer $\Rightarrow (y,x)\in R$. So, $R$ is symmetric.
For transitivity, let $(x,y)\in R$ and $(y,z)\in R$, so $x-y=$ integer and $y-z=$ integers, then $x-z$ is also an integer.
$\therefore (x,z)\in R$. So, $R$ is transitive.