8. Define relations $R$ and $S$ on $\mathbb{N} = \{0,1,2,\ldots\}$ by (1) $xRy$ if and only if $x \equiv y \pmod{7}$, and (2) $xSy$ if and only if $x \equiv y \pmod{13}$. For each of the following relations on $\mathbb{N}$, determine whether it is an equivalence relation. If it is, give a proof. If it is not, explain which property of an equivalence relation fails. In each case, justify your answer. Answers without justification will receive no points.

(a) (4 points) $R \cap S$

(b) (4 points) $R \cup S$