Mathematics Question from JEE Main 2023 · 11 April, Shift 2

For $a \in \mathbb{C}$, let:

$$A = \{ z \in \mathbb{C} : \operatorname{Re}(a + \overline{z}) > \operatorname{Im}(\overline{a} + z) \}, \text{and} \quad B = \{ z \in \mathbb{C} : \operatorname{Re}(a + \overline{z}) < \operatorname{Im}(\overline{a} + z) \}.$$

Among the two statements:

(S1): If $\operatorname{Re}(a), \operatorname{Im}(a) > 0$, then the set $A$ contains all the real numbers.

(S2): If $\operatorname{Re}(a), \operatorname{Im}(a) < 0$, then the set $B$ contains all the real numbers.