Question

A relation R on the set of complex numbers is defined by $z_{1}R{z}_2$ if and only if $\frac{z_1-z_2}{z_1+z_2}$ is real. Show that R is atransitive.

Answer 1

$\frac{z_1-z_2}{z_1+z_2}$ is real
$\therefore \frac{(x_1+i{y}_1)-(x_2+i{y}_2)}{(x_1+i{y}_1)+(x_2+i{y}_2)}=\frac{(x_1-x_2)+i(y_1-y_2)}{(x_1-x_2)-i(y_1-y_2)}$
Multiply above and below by conjugate of denominator and it will be real if the imaginary part of numerator is zero because denominator being
$z\overline{z}=|z{|}^2$ is real
This will give $x_1{y}_2-x_2{y}_1=0$
similarly $z_2{R}_{z3}\Rightarrow \frac{x_2}{y_2}=\frac{x_3}{y_3}$
Hence it follows that $\frac{x_1}{y_1}=\frac{x_3}{y_3}\therefore z_1{R}_{z3}$
Hence the relation R is transitive.