As part of a HW assignment in the course elementary set theory, I was given the following question:
Let $A$ be a set and let $T$ and $S$ be two equivalence relations on $A$.
Prove: $S\circ T\subset S\cup T \Longrightarrow S\cup T$ is an equivalence relation.
I managed to prove reflexiveness and symmetry of $S\cup T$, but when i try to prove transitivity I have the following problem: I assume $(a,b),(b,c)\in S\cup T$ and I split into four cases (each pair belongs to $S$ or $T$) but in the case that $(a,b)\in S , (b,c)\in T$ ,I can't reach $(a,c)\in S\cup T$
help please?
Hint. Can you handle the case $(c,b)\in T, (b,a)\in S$?