Often, the easiest and shortest way for both the writer and the reader is to write things down formally.

Suppose is a partial order on . We need to prove that is a strict partial order.

Irreflexivity. Let . Then , so . That was easy.

Transitivity. Suppose , i.e., , and . Then . It's left to show that . Suppose ; then by antisymmetry of R we have x = y, a contradiction.

Thinking back, it is right that we had to use antisymmetry. If every reflexive and transitive (but not necessarily antisymmetric) relation would produce a strict partial order when one subtracts , then it is unlikely that joining a strict order with would produce an antisymmetric relation as the second part of the problem says.

I recommend similarly writing out the second part in every detail.