Chapter 1.7, Problem 4E

Let $R$ be the relation “congruence modulo 5” defined on $Z$ as follows: $x$ is congruent to $y$ modulo $5$ if and only if $x-y$ is a multiple of $5$, and we write $x\equiv y\left(\text{mod}5\right)$.

a. Prove that “congruence modulo $5$

” is an equivalence relation.

b. List five members of each of the equivalence classes $\left[0\right],\left[1\right],\left[2\right],\left[8\right],$ and $\left[-4\right]$.