I was given the following task to do
Which of the following relations are partially ordered relations and which are linearly ordered relations.
$ (x, y) \preceq_1 (x', y') $ if both $x \le x'$ and $y \le y'$ are true.
$ (x, y) \preceq_2 (x', y') $ if either $x \le x'$ and $y \le y'$ is true.
Could you please explain how do we solve such a problem ?
The first relation is a partial order but not total, since $(1,2)$ and $(2,1)$ are not comparable.
The second relation is not a partial order, since $(1,2)\prec_2(2,1)$ and $(2,1)\prec_2(1,2)$, but $(1,2)\ne (2,1)$. Antisymmetry fails.