Let $σ$ be the relation on $R×R$ in which $(a,b) σ (x,y)$ if and only if $a ≤ x$ and $b ≤ y$. Prove that $σ$ is a partial order relation.

Question

I can prove it is reflexive and transitive but I cannot prove that it is antisymmetric. Help.

Answer 1

$$(a,b) \sigma (x,y) \implies a \leq x \wedge b \leq y$$

$$(x,y) \sigma (a,b) \implies x \leq a \wedge y \leq b$$

If $a \leq x$ and $x \leq a$ what can you conclude?

what about $b\leq y$ and $y \leq b$?