I would appreciate any help available for the following problem:
Let $S$ be a set. Let $T$ be the set of all relations on $S$. Construct a relation $\equiv$ on $T$ in the following way: for $\sim, \approx \in T$, say $\sim \equiv \approx$ if $\forall s,s' \in S, s \sim s' \implies s\approx s'$. Determine whether $\equiv$ is an equivalence relation on $T$, and if so, what its equivalence classes are.
What I did was this, however, I don't think it is correct.
Define a relation $@ \in T$ in the following way $\forall x,y \in S$, $x@y$ is not true, thus no elements in $S$ are related via $@$. The relation $\equiv$ is not an equivalence relation on T because it's not symmetric. Because if we let $@$ be the relation described, and let $@$ be the relation defined so that $\forall x,y \in S, x@y$ is true. Vacuously true statements would give $@ \equiv @$. But this is where I think I am going wrong because I don't know if S is a nonempty set?
HINT:
Let $R$ be the relation such that for every $x,y\in S$, $x\mathrel{R}y$. That is $R=S\times S$.
Show that every $\sim\in T$ has the property, $\sim\equiv\mathrel{R}$.