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?

  • 2
    $\begingroup$ You're not missing anything. Sometimes you get an easy problem. $\endgroup$ – John Douma Jun 5 '13 at 15:28
  • $\begingroup$ Clearly, you just showed that $\leq$ defined above is a preordering on $\mathbb{C}$. Easy one:) $\endgroup$ – Bartek Pawlik Jun 5 '13 at 15:29
  • $\begingroup$ I honestly can not remember the last time I had an easy problem - hence my suspicion. Thank you. $\endgroup$ – Wortel Jun 5 '13 at 15:29

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