Is this sentence symbolication correct? Predicate logic

Question

Key: B_ = _lives in Bangladesh C__ = cooks for d = Donald

My sentence: Donald cooks for all but only those inhabitants of Bangladesh who do not cook for themselves.

My symbolization: $\forall x((Bx\to \forall y(Cyx)) \iff Cdx)$

Is this correct? The sentence is quite confusing. Thanks! :)

Answer 1

Hints: You haven't captured the negation: to express $y$ does not cook for themself, is $\lnot Dyy$. $\forall y(Cyx)$ means that everybody cooks for $x$.

You should also think about people who do not live in Bangladesh: with your statement, Donald is very busy indeed: he cooks for everybody who does not live in Bangladesh. The natural language statement doesn't say anything about people who do not live in Bangladesh. So the overall structure should be $\forall x (Bx \to (\ldots \iff \ldots))$ not $\forall x ((Bx \to \ldots) \iff \ldots)$.

Answer 2

Donald cooks for all but only those inhabitants of Bangladesh who do not cook for themselves.

No; there is no biconditional here.   Say: "Everyone, except inhabitants of Bangladesh who cook for themselves, will have Donald cook for them."