Is the partial order relation a total order relation

Question

Let $T$ be the partial order relation defined on $\mathbb{N}\times \mathbb{N}$ by $(a,b)\, T\, (c,d)$ if and only if $a$ is smaller than or equal to $c$ and $b$ is smaller than or equal to $d$. Is it a total order relation?

I'm thinking it's yes but I don't know how to justify?

Answer 1

Take out a piece of paper, and draw the first few points of $\Bbb N\times \Bbb N$ in a square, say up to $5\times 5$ or $6\times 6$. Now colour every point $(a,b)$ such that either $(a,b)T(3,3)$, or $(3,3)T(a,b)$. Can you see that some points are uncoloured, i.e. incomparable to $(3,3)$? What is the conclusion?

Answer 2

It is not a total order relation because you cannot compare $(1,2)$ and $(2,1)$.

This means that you cannot say $(1,2)\, T\, (2,1)$ or $(2,1)\, T \,(1,2)$.