If $xR_1y$ if $x^2 + y^2$ is divisible by $5$, is the relation transitive?

Question

Been asked to prove if this is/isn't transitive. Not sure where to start on if it is transitive or not. Also by proving it is/isn't, do I simply need to give an example where it is/isn't true?

Thanks!

Answer 1

$R$ is transitive if, whenever $x R y$ and $y R z$, we have $x R z$. Remembering what the relation means, the question you need to answer is: whenever $x^2 + y^2$ is divisible by 5 and $y^2 + z^2$ is divisible by 5, is $x^2 + z^2$ divisible by 5?

To prove the relationship is transitive, you need to show this is always true. To prove it is not transitive, you need to find $x, y, z$ such that it's not true - that is, with $x^2 + y^2$ divisible by 5, $y^2 + z^2$ divisible by 5, and $x^2 + z^2$ not divisible by 5.

Answer 2

To prove it isn't true you need one example where it fails but to prove it you have to show it is true for all possible pairs, which means you need algebra.

Play around with some examples of xRy and yRz.