Equivalence relations and classes

Question

$T$ is defined on $\mathbb{N} \times \mathbb{N}$ by $(a,b)\mathrel{T}(c,d)$ if and only if $a \leq c$ and $b \leq d$.

I know this is a partial order relation as it is Transitive, Anti Symmetric and Reflexive but I'm not sure if its total order relation and whether they are well order relations?

try compare $(1,2)$ and $(2,1)$. – Ofir Schnabel Apr 29 '15 at 08:12 — Ofir Schnabel, Apr 29 '15 at 08:12

score 1 · Answer 1 · answered Apr 29 '15 at 08:14

1

It is not a total order: just compare $(1,0)$ and $(0,1)$. It is a well partial order, a result known as Dickson's lemma.

answered Apr 29 '15 at 08:14

J.-E. Pin

40,163

Im assuming a well order relation is different from a well partial order? As I don't think we have learnt Dickson's Lemma yet – Sunny Apr 29 '15 at 08:20
@Sunny. Look at the first sentence of this page on Well orders : In mathematics, a well-order relation (or well-ordering)... As a rule, please look at Wikipedia before asking any question on math.stackexchange. – J.-E. Pin Apr 29 '15 at 08:25

Equivalence relations and classes

1 Answers1