Partial Order Relation help

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

Suppose $R$ is a partial order on $a$ . We need to prove that $R\setminus I_a$ is a strict partial order.

Irreflexivity. Let $x\in a$ . Then $(x,x)\in I_a$ , so $(x,x)\notin R\setminus I_a$ . That was easy.

Transitivity. Suppose $(x,y),(y,z)\in R\setminus I_a$ , i.e., $(x,y),(y,z)\in R$ , $x\ne y$ and $y\ne z$ . Then $(x,z)\in R$ . It's left to show that $x\ne z$ . Suppose $x=z$ ; 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 $I_a$ , then it is unlikely that joining a strict order with $I_a$ would produce an antisymmetric relation as the second part of the problem says.

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