How do you determine if a relation is transitive?

Question

Suppose I have the relation P such that $$ x P y $$ iff

$$ x = y^2 $$

How do I determine whether or not the relation is transitive?

Answer 1

It is not transitive because $16P4$ and $4P2$ hold, while $16 P 2$ is false.

Answer 2

A relation $R$ over a set $X$ is said to be transitive if, for all $a, b, c \in X$, $aRb$ and $bRc$ $\Rightarrow$ $aRc$.

For your relation $P$, taking $x = y^2$ and $y = z^2$, we see that $x = (z^2)^2 \neq z^2$.

Thus, $xPy$ and $yPz$ does not imply $xPz$, and $P$ is not transitive.