0

I'm stuck on how to prove this, most likely because I haven't really done an example like this before and I'm pretty new to partial order relations. It would be greatly appreciated if you could help me out, because I'm stressing over this, so any help would greatly be appreciated, thanks.

Question

Let $\phi$ be the relation on $\mathbb{R}\times \mathbb{R}$ in which $(a,b)\phi (x,y) \iff a\leq x$ and $b\leq y$. Prove that $\phi$ is a partial order relation. Additionally, is $\phi$ a total order relation? Explain why/why not.

James Blair
  • 367
  • You need to prove reflexivity, transitivity and antisymmetry. Its not a total order. Give a counterexample. – Wuestenfux May 22 '17 at 09:03
  • Showing reflexivity, transitivity and antisymmetry should be straightforward. Finding a counterexample often needs more practice, so I'll give one: $p=(0, 1)$ and $q=(1, 0)$. These two pairs neither satisfy $p \phi q$, $q \phi p$ nor $p = q$. – md2perpe May 22 '17 at 10:33

0 Answers0