I am given a definition which states that a 'preodering on a set is a relation that is reflexive and transitive.'
Show that a relation $\leq$ defined on $\mathbb{C}$ by $z_1 \leq z_2$ iff $|z_1| = |z_2|$ is a preodering on $\mathbb{C}$.
I must be missing something here because clearly for any $z \in \mathbb{C}$ we have $|z| = |z|$ and if $z_1 \leq z_2$ and $z_2 \leq z_3$ then that implies that $|z_1| = |z_2| = |z_3|$ which means that $z_1 \leq z_3$. I must be reading the question wrong surely. What am I missing?