For a ∈ ℂ , let A = z ∈ ℂ: Re ( a + ¯ z ) > Im ( ¯ a + z ) and B =...

Let $a = {x}_{1} + i{y}_{1}, \overset{¯}{a} = {x}_{1} - i{y}_{1}$

And $z = {x}_{2} + i{y}_{2}, \overset{¯}{z} = {x}_{2} - i{y}_{2}$

Now,

$a + \overset{¯}{z} = \left({x}_{1} + {x}_{2}\right) + i\left({y}_{1} - {y}_{2}\right)$

and $\overset{¯}{a} + z = \left({x}_{1} + {x}_{2}\right) + i\left({y}_{2} - {y}_{1}\right)$

Now according to the question,

$Re ( a + \overset{¯}{z} ) = {x}_{1} + {x}_{2}$ and

$Im ( \overset{¯}{a} + z ) = {y}_{2} - {y}_{1}$

$A = \left\{z: {x}_{1} + {x}_{2} > {y}_{2} - {y}_{1}\right\} = \left\{z: {x}_{1} + {y}_{1} + {x}_{2} > {y}_{2}\right\}$
$B = \left\{z: {x}_{1} + {x}_{2} < {y}_{2} - {y}_{1}\right\} = \left\{z: {x}_{1} + {y}_{1} + {x}_{2} < {y}_{2}\right\}$

If ${y}_{2} = 0$ and ${x}_{1}, {y}_{1} > 0$ then

$A = { z: {x}_{2} > - \left({x}_{1} + {y}_{1}\right) }$

A covers a part of negative real axis and therefore, does not contain whole real axis

Similarly if ${y}_{2} = 0$, and ${x}_{1}, {y}_{1} < 0$, then
$B = { z: {x}_{2} < - \left({x}_{1} + {y}_{1}\right) }$
$∴$ $B$ covers part of positive real axis and therefore does not cover whole real axis.

Hence, both are false.