For every partial order ≤ is the relation < transitive?

Question

For every general partial order ≤ is the relation < := ≤ ∩ ≠ transitive

I tried working with the definition of the partial order. A partial order is antisymmetric, transitive and reflexive. The intersection of two sets is the set of the elements that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

So the question is now, if the transitivity depends on the equality of the elements, and therefore the reflexitivity?

My guess would be that the statement can be true, but doesn't have to.

But I couldn't come up with a good counter example.

Thanks in advance for the help!

Answer 1

Hint: If $a<b$ and $b<c$ then $a\leq b$ and $b\leq c$. So you know $a\leq c$. Now you just need to show $a\neq c$. What happens if $a=c$?