Which of the following relations are partially ordered relations and which are linearly ordered relations.

Question

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 ?

score 1 · Accepted Answer · answered Sep 29 '20 at 07:08

1

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.

answered Sep 29 '20 at 07:08

Wuestenfux

20,964

What is (x, y). Set, ordered pair, or just an arbitrary object ? – Rustem Sadykov Sep 30 '20 at 06:03
1

Its an ordered pair. – Wuestenfux Sep 30 '20 at 06:56

Which of the following relations are partially ordered relations and which are linearly ordered relations.

1 Answers1