Prove that there is no relationship $\prec$ in $\mathbb C$ such that:
a) Given $z,w \in \mathbb C$, one and only one of the following conditions is valid:$z=w, z \prec w, w \prec z$.
b)If $z_1 \prec z_2$ and $z_3 \in \mathbb C$, then $z_1+z_3 \prec z_2+z_3$.
c)If $z_1 \prec z_2$ and $0 \prec z_3$, then $z_1z_3 \prec z_2z_3$.
a) $z = a + bi, w = c + di$
Case 1 $z=w \to a=c, bi=di$ so $z \prec w$ or $w \prec z$ are false.
Case 2 $z \prec w \to bi \lt di $ or $ bi = di$ and $a \lt c$
Case 3 $w \prec z \to di \lt bi $ or $ bi = di$ and $c \lt a$ so if $z=w$ case 2 and 3 are impossible, if $z \prec w$ case 1 and 3 are impossible and if $w \prec z$ case 1 and 2 are impossible.
b) $z_1 = a + bi, z_2 = c + di, z_3 = e + fi$ if $z_1 \prec z_2$ then $bi \lt di$ or $di = fi$ and $a \lt c$ so $z_1 = (a + bi) + z_3 = (e + fi) \prec z_2 = (c + di) + z_3 = (e + fi) \to (a + bi) + (e + fi) \prec (c + di) + (e + fi) = (a+e) + i(b+f) \prec (c+e)+i(d+f)$ but $a+e \lt c+e$ and $b+f \lt d+f $ therefore $z_1 + z_3 \prec z_2 + z_3$.
c) $z_1 = a + bi, z_2 = c + di, z_3 = e + fi$ if $z_1 \prec z_2$ and $ 0 \prec z_3 $ then $z_1 = (a + bi) * z_3 = (e + fi) \prec z_2 = (c + di) * z_3 = (e + fi) \to (a + bi) * (e + fi) \prec (c + di) * (e + fi) = a*e + a*fi +bi*e + bi*fi \prec c*e + c*fi +di*e + di*fi = (a*e - b*f) + i(a*f +b*e) \prec (c*e - d*f) + i(c*f +d*e)$ but $ a*e - b*f \lt c*e - d*f$ and $ a*f +b*e \lt c*f +d*e$ therefore $z_1z_3 \prec z_2z_3$.
Is my answer correct?
Grateful for the attention.
