For two binary relations R, S on a set X, we define their composition:

I have proven two statements on this and am not sure about the third one. They're related so I'll post them all.

1.If R and S are commuting transitive relations on X, then their composition is also transitive.

Proof.Let and for some Then there exist such that

Then we have and since we also have . There must therefore exist such that

But then, from the transitivity of R and S, we get

which means that

If R, S are preorders and their composition is transitive, then they must commute.

2.

Proof.Let and Then there is such that

Let's take

Then, since R and S are reflexive, we have

(as we had in the previous proof). Therefore and and by transitivity But and so we are done.

3.If If R, S are transitive and their composition is transitive, then they must commute.

I believe this one is false, because the above argument fails to work for it, but that's no proof. I can't think of a way of refuting this statement other than by looking at random examples and hoping for the best. Could you suggest something more systematic? I would also find it very kind if someone would tell me about any embarrassing mistakes I made in my proofs.