Chapter 1.7, Problem 5E

Let $R$ be the relation “congruence modulo $7$

” defined on $Z$ as follows: $x$ is congruent to $y$ modulo $7$ if and only if $x-y$ is a multiple of $7$, we write $x\equiv y\left(\text{mod7}\right)$.

a. Prove that “congruence modulo $7$

” is an equivalence relation.

b. List five members of each of the equivalence classes $\left[0\right],\left[1\right],\left[3\right],\left[9\right],$ and $\left[-2\right]$.