Let $R$ be a relation on the set of real numbers $\mathbb{R}$ defined by $xRy$ if and only if $x-y\in \mathbb{Z}$. Which of the following is true about $R$?

Solution

1. A relation $R$ is reflexive if $x-x=0$, which is an integer for all $x\in \mathbb{R}$.
2. It is symmetric because if $x-y=k$ for some $k\in \mathbb{Z}$, then $y-x=-k$, which is also an integer.
3. It is transitive because if $x-y=k_1$ and $y-z=k_2$ for integers $k_1,k_2$, then $$\begin{array}{rl}x-z& =(x-y)+(y-z)\\ & =k_1+k_2\end{array}$$, which is an integer.
4. Since $R$ is reflexive, symmetric, and transitive, it is an equivalence relation.
5. Hence the answer is (B).