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

Question

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

Accepted Answer

Take $a=2+10i$ (satisfying $\operatorname{Re}(a),\operatorname{Im}(a)>0$) and test a real $z=-12$: we need $\operatorname{Re}(a+z)>\operatorname{Im}(a+z)$, i.e. $2-12>10$, i.e. $-10>10$ --- false. So (S1) fails for this real number, meaning (S1) is false in general. Similarly, taking $a=-2-10i$ and $z=12$: we need $\operatorname{Re}(a+z)<\operatorname{Im}(a+z)$, i.e. $-2+12<-10$, i.e. $10<-10$ --- also false. So (S2) is false too. Both statements are false.

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