Let ~ be the quasi-order relation. and let // be defined as the relation s~t and t~s(s,t elements of S). Show that ~ induces a partially ordered relation on the set of equivalence classes relation //, denoted S/(//) "quotient set"
I haven't had much experience with set theory and need help with what I need to actually show here.
$\sim$ is a preorder or quasiorder, so we know that it's reflexive and transitive.
$$x \sim x$$ $$x \sim y ∧ y \sim z → x \sim z$$
To show that $\sim$ is a partial order on $//$ we need the facts above and we need to prove that it's antisymmetric (on $//$).
$$x \sim y ∧ y \sim x → x \: // \: y$$
But this is a trivial consequence of the definition $x \: // \: y ↔ x \sim y ∧ y \sim x$.