How do I determine whether this relation is transitive?

Question

I've been given this relation, and I'm supposed to determine whether it is transitive. I understand the definition of transitive (sort of, in theory) but I'm not sure how to put it in action here. $$\{(x,y) \in \ \mathbb N^2 \ | \ x+y=10\}.$$

Answer 1

A relation $R$ is said to be transitive if for all $a,b,c$ in its domain,

$aRb$ and $bRc \Rightarrow aRc$

In your case, the relation would be transitive if, for all $x,y,z \in \mathbb N$, $x+y=10$ and $y+z=10$ implied $x+z=10$, which is clearly not the case.