Proving Equivalence Relations On A Set

Question

Let X be the set of all nonempty subsets of {1,2,3}. Then

X = {{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

Define a relation R on X as follows: for all S and T in X, SRT if, and only if, the least element of S equals the least element of T. Prove that R is an equivalence relation on X.

So my question is, what are S and T? Are they some sort of set or are they elements in X? How would I go about proving that R is an equivalence relation on X?

Answer 1

You have to prove that $R$ is reflexive, symmetric and transitive.

1)Reflexive:

Clear since if we have the same set then least elements are equal. So, $SRS$.

2) Symmetric:

Suppose $SRT$. Then least element of $S$ equals least element of $T$. Hence, least element of $T$ equals least element of $S$. So, $TRS$.

3) Transitive:

Suppose $SRT$ and $TRU$. Then least element of $S$ equals least element of $T$ and least element of $T$ equals least element of $U$. Hence, least element of $S$ equals least element of $U$. So, $SRU$.