Union of equivalence classes

Question

Given two equivalence relations $R,T$ on a set $S$ with $R \subseteq T$. There are two quotient sets induced by each relation, $S/R = \{S_{Rx}:x\in S\}$, and $S/T=\{S_{Tx}:x\in S\}$. How would I go about showing that, $$S_{Tx}=\bigcup_{y\in S_{Tx}}S_{Ry}$$

Answer 1

Let x/R be the equivalent class from the equivalence
relation R that contains x.
Assume R,T binary relations for S and R subset T.

a/T = $\cup${ x/R | x in a/T }. Proof:

If x in a/T, then x in a/R whence x is in the union.
Assume x is in the union.
Then some b in a/T with x in b/R.
As R subset T, xRb, xTb, bTa, xTa. Thus x in a/T.