Confusion on Propositional Logic Question

Question

I have this proposition:$(\forall a\in \Re )(\forall b \in\Re)(e^{a+bi}=2i\Rightarrow a=ln2,b=\frac{\pi}{2})$ Where I am asked to find what is false about this, 2 things came to mind, the initial part ofthe proposition states that that $a$ and $b$ are said to be some arbitrary value and we are then told $a$ and $b$ have this explicit value, second of all theta isn't given as $\frac{\Pi}{2}+2\Pi n$. The hint was that we had to introduce a quantifier after $\Rightarrow$ regarding $n$ being part of the real numbers.

Answer 1

I think you've basically got the flaw in the statement, it's of course that if $b=\pi/2+2n\pi$ then $e^{a+ib}=2i$. So you can't conclude that $b=\pi/2$ from that. Your second comment indicates you're clear about that.

The quantifiers and the statement that $a$ and $b$ has specific values is not in itself a flaw. The statement says that for any $a$ and $b$ fulfilling $e^{a+ib}=2i$ you would have $a=\ln 2$ and $b=\pi/2$. It doesn't outright say that they have that value, if for example $a=b=0$ then you have that $e^{a+ib}=2i$ is false so for the implication to hold the statement $a=\ln 2$ and $b=\pi/2$ is allowed to be false (or true).

In fact the negation of the statement is that there exists $a$ and $b$ such that $a\ne\ln2$ or $b\ne\pi/2$ and still $e^{a+ib}=2i$. For example $b=5\pi/2$ and $a=\ln2$ would prove that the negation of the statement is true.