Let $R$ and $S$ be two partial orders on a set $X$, and $T$ is a relation on $X$ such that $aTb$(i.e. $a,b$ ∊ X) if and only if both $aRb$ and $aSb$ hold. Is $T$ also a partial order on $X$?
Attempt
I know I need to prove T is reflective, antisymmetric, and transitive. I succeeded to prove T is reflective and transitive but have no idea about how to prove T is antisymmetric. I only know $aTb$ and $bTa$ hold where $a=b$, also $aSb$ and $bSa$.
I suppose there are $aTb$ and $bTa$, but with the proofs I have now, I cannot get $a=b$ when $aTb$ and $bTa$ hold. What should I do next?