# Doubt pertaining to this Equivalence Relation.

$1$. 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.

• That is the definition of an equivalence relation, but I don't see how it's relevant or how the given statement follows from it. – anomaly Sep 8 '14 at 4:42
• False. Take the set $\mathbb{Z}$ of all integers, and consider the universal relation (that relates every number with every other number). Then there is only one equivalent class, but the set $\mathbb{Z}$ itself is infinite. – E W H Lee Sep 8 '14 at 4:45
• thanks....can provides some more example ,so that i exactly understand on how to model this question – anish Sep 8 '14 at 5:00
• What you think is false and the reason why you think that it is true, is also false. – Hirak Sep 8 '14 at 5:23

## 2 Answers

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 $,,$ i.e the remainders when divided by $3$ But $\mathbb Z$ is not finite.

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.