$\ A+A=\{3i,3+2i,3+4i\}$. find A

Question

Can someone help me and answer this exercise of complex analysis? I don't know how to try to resolve this because I have never seen an exercise like this.

Answer 1

Hint Let $z = a+bi \in A$. Then $z+z \in A+A=\{3i,3+2i,3+4i\}$. This gives you 3 choices for $z$.

Since $A$ must have at least two elements, this gives you 4 possible choices for $A$, and you simply check which ones work.

You can reduce the case by case analysis by showing that $A$ has exactly two elements, and showing that these correspond to $z+z$ having the smallest/largest imaginary part.