Show that a binary relation ($a \le b$) or ($a = b$ and $c \leq d $) is an order relation

Question

I have to show that if there are $a + bi$, $c + di$ where $a,b,c,d\in C$ and $i=\sqrt(-1)$ and also

$a + bi \preceq c + di \iff $ $(a <b)$ or ($a=b$ and $c \leq d$)

the $ \preceq$ is an order relation in $C$.

How can we show antisymmetry, transitivity and reflexivity?

For the reflexivity, I think that we could say that b cannot be less than a, so, the $a \le b$. Since $a=b$ and $a \le b \iff a + bi \preceq a + bi $ What about the other two (antisymmetry and transitivity) ?

Answer 1

Surely the relation is not even reflexive ?

Reflexivity requires that $z \preceq z \space \forall z \in \mathbb{C}$. But if $a>b$ then the conditions on the right hand side of the relation are not met so $a+bi \not \preceq c+di$. Hence, for example, $2+i \not \preceq 2+i$.