Proposition. Let be a relation on Z where for all a; b $2$ Z$,$ a b if and only if $.$a C $2$b$/0.$mod $3/.$ The relation is an equivalence relation on Z$.$ Proof. Assume a a$.$ Then $.$a C $2$a$/0.$mod $3/$ since $\mathrm{.}3$a$/0.$mod $3/.$ Therefore, is reflexive on Z$.$ In addition, if a b$,$ then $.$a C $2$b$/0.$mod $3/,$ and if we multiply both sides of this congruence by $2,$ we get $2.$a C $2$b$/20.$mod $3/\mathrm{.}2$a C $4$b$/0.$mod $3/\mathrm{.}2$a C b$/0.$mod $3/.$b C $2$a$/0.$mod $3/$ : This means that b a and hence, is symmetric. We now assume that.a C $2$b$/0.$mod $3/$ and $.$b C $2$c$/0.$mod $3/.$ By adding the corresponding sides of these two congruences, we obtain $.$a C $2$b$/$ C $.$b C $2$c$/0$ C $0.$mod $3/.$a C $3$b C $2$c$/0.$mod $3/.$a C $2$c$/0.$mod $3/$ : Hence, the relation is transitive and we have proved that is an equivalence relation on Z$.$