Let $R$ and $S$ be equivalence relations on set $A$.
Prove that if relation $SR$ is symmetric, then $SR = RS$.
$R$ and $S$ are equivalence relations on set $A$, so $R\subseteq A\times A$ and $S\subseteq A\times A$. In order to be equivalence relations, they need to meet these three conditions:
- be reflexive: $a\in A : aRa$ (analogically for $S$)
- be symmetric: $a,b\in A : aRb \rightarrow bRa$
- be transitive: $a,b,c\in A : (aRb \wedge bRc)\rightarrow aRc$
I do not really know where to start, I was thinking about it the following way:
Composition of relations $S$ and $R$ can be written as $SR\subseteq A\times A$, as both relations are on set $A$. If relations $S$ and $R$ are equivalence relations, then their composition should also be an equivalence relation.
but I am not sure whether it is correct and I also get stuck there. It is kind of problematic, as I need a formal proof to it.