$1$. True or false? If $R$ is an equivalence relation on a set $S$ and it has only finitely many equivalence classes altogether, then $S$ itself is a finite set.
I think the answer is true because if $R$ is an equivalence relation then it will be reflexive,symmetric and transitive.
On
Let $S$ be any set. Consider any partition of $S$ into subsets. That partition determines an equivalence relation on $S$, namely, two elements of $S$ are equivalent if and only if they are in the same subset. So there are no restrictions on the size of $S$, nor on the number of equivalence classes, except that the latter can't be any bigger than the former.
On the set of integers $\mathbb Z$ define a relation $\rho$ by $a\rho b$ iff $3|a-b$ Then $\rho$ is equivalence relation and eqivalence classes are $[0],[1],[2]$ i.e the remainders when divided by $3$ But $\mathbb Z$ is not finite.