Checking whether a relation is transitive I'm currently preparing for my maths exam and one of the questions is to check whether or not a relation is transitive or not. I'm unsure on how to check if there is a transitive relation given an equation.
The questions I would be getting is like this.
Let $Q$ denote the relation on set $\mathbb Z$ of integers, where integers $x$ and $y$ satisfy $xQy$ if and only if
$$x - y = x^2 + y^2 - 2xy$$
I know how to figure out if it is symmetric and reflexive but I'm unsure on how to check the transitive part. Any help is appreciated, thank you.
 A: If the question is "Is this relationship transitive?" then you need a little bit of gut feeling to get started. If you suppose that it is transitive, then you'll need a general proof. If you think it's not transitive, then one example will suffice.
As noted by Noy Soffer, $x^2 + y^2 - 2xy = (x - y)^2$. We thus have $xQy$ if and only if $x - y = (x - y)^2$. For this, it is sufficient that $x = y$ (hence the relation is reflexive, as you probably already noticed). If $ x \neq y$, then we can divide both sides of the equation by $x - y$ and get $ 1 = x - y$. We conclude that $xQy$ means that one of the following hold
$$
x - y = 1 \text{ or } x = y
$$
That's just the basic analysis. Now what's your intuition? If the relationship would be transitive, then if $x - y = 1$ and $y - z = 1$, we would have to conclude that either $x - z = 1$ or that $x = z$. Can you find examples of $x, y$ and $z$ for which $x - y = 1$, $y - z = 1$, but neither $x = z$ nor $x - z = 1$? If so, then you've proven it's not transitive. If you can't seem to find counterexamples, then maybe the relationship is transitive. In that case, try proving that $x - y = 1$ and $y - z=1$ implies either $x - z =1$ or $x = z$. Let me know what you find.
