For $\mathrm{a}\in \mathrm{ℂ}$, let $\text{A}=\{\mathrm{z}\in \mathrm{ℂ}:\mathrm{Re}(\mathrm{a}+\overline{\mathrm{z}})>\mathrm{Im}(\overline{\mathrm{a}}+\mathrm{z}\left)\right\}$ and $\text{B}=\{\mathrm{z}\in \mathrm{ℂ}:\mathrm{Re}(\mathrm{a}+\overline{\mathrm{z}})<\mathrm{Im}(\overline{\mathrm{a}}+\mathrm{z}\left)\right\}$. Then among the two statements:
(S1) : If $\mathrm{Re}\left(\mathrm{a}\right),\mathrm{Im}\left(\mathrm{a}\right)>0$ , then the set   contains all the real numbers
(S2) : If  $\mathrm{Re}\left(\mathrm{a}\right),\mathrm{Im}\left(\mathrm{a}\right)<0$, then the set B contains all the real numbers, 
 

Let $\mathrm{a}={\mathrm{a}}_1+{\mathrm{ia}}_2,\mathrm{z}=\mathrm{x}+\mathrm{iy}$; $\mathrm{A}=\{\mathrm{x}+\mathrm{iy}:\mathrm{Re}(\mathrm{a}+\overline{\mathrm{z}})>{\mathrm{I}}_{\mathrm{m}}(\overline{\mathrm{a}}+\mathrm{z})\}$

$a_1+x>-a_2+y$
$$\begin{gathered} {\mathrm{a}}_1+{\mathrm{a}}_2>\mathrm{y}-\mathrm{x} \\ \mathrm{S}-\mathrm{I}:{\mathrm{a}}_1>0,{\mathrm{a}}_2>0, \end{gathered}$$

for all real numbers, y = 0
$-\mathrm{x}<{\mathrm{a}}_1+{\mathrm{a}}_2$ is not true for all real x, for given ${\mathrm{a}}_1,{\mathrm{a}}_2$ 
$\therefore \mathrm{S}-\mathrm{I}$ is false 

$a_1+x<-a_2+y$

${\mathrm{a}}_1+{\mathrm{a}}_2<\mathrm{y}-\mathrm{x}$
Similarly $$\begin{gathered} {\mathrm{a}}_1<0,{\mathrm{a}}_2<0, \\ -\mathrm{x}>{\mathrm{a}}_1+{\mathrm{a}}_2, \end{gathered}$$

for all real x is also not true
$\therefore \mathrm{S}-\mathrm{II}$ is false