(a)
$$ z=Re(z)+Im(z)\cdot i, \bar(z)=Re(z)-Im(z)\cdot i $$ $$ \Rightarrow z+\bar{z}=Re(z)+Im(z)\cdot i+Re(z)-Im(z)\cdot i=2Re(z) $$
(b)
$$ z=Re(z)+Im(z)\cdot i, \bar(z)=Re(z)-Im(z)\cdot i $$ $$ \Rightarrow z-\overline{z}=Re(z)+Im(z)\cdot i-(Re(z)-Im(z)\cdot i) $$ $$ =Re(z)+Im(z)\cdot i-Re(z)+Im(z)\cdot i=2Im(z)\cdot i $$
(c)
$$ z=Re(z)+Im(z)\cdot i, \bar(z)=Re(z)-Im(z)\cdot i, |z|=\sqrt{Re(z)^2+Im(z)^2} $$ $$ \Rightarrow z\cdot \bar{z}=(Re(z)+Im(z)\cdot i)\cdot (Re(z)-Im(z)\cdot i) $$ $$ =Re(z)^2-Re(z)Im(z)\cdot i+Re(z)Im(z)\cdot i-Im(z)^2\cdot i^2 $$ $$ = Re(z)^2-Im(z)^2\cdot (-1)=Re(z)^2+Im(z)^2=|z|^2 $$