how to translate this sentence into predicates and quantifier?

Question

I need to translate the following sentence:

All women fear close-minded men. 

F(x): x is female;
M(x): x is male;
O(x): x is open mind
S(x,y): x is scared of y 
The Universe of x and y are people. 

My answer is :

∀x∀y [F(x)∧M(y)∧¬O(y)∧S(x,y)]       

meaning all women fears all the men that are close minded.

But my teacher says the answer is :

∀x∃y [F(x)∧M(y)∧¬O(y)∧S(x,y)]

But my understanding for the above translation is that: all women fears some men who are close minded.

Do you think my answer is right? what is your answer?

Answer 1

Neither is correct.

You say: everyone is a woman and everyone is a close minded man and everyone scares everyone.$$\forall x~\forall y \big[F(x)\wedge M(y)\wedge \neg O(y)∧S(x,y)\big]$$

Your teacher says: everyone is a woman though someone is a close minded man whom scares everyone.$$\forall x~\exists y~\big[F(x)\wedge M(y)\wedge \neg O(y)\wedge S(x,y)\big]$$

You want to say: If anyone is a woman, and if anyone is a close-minded man, then the later will scare the former.

$$\forall x~\forall y~\Big(\big(F(x)\wedge M(y)\wedge\neg O(y)\big)\to S(y,x)\Big)$$

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Your domain is implicitly "people".   If you want to restrict a universal quantifier to a category, you must use the conditional connective.

$\forall z~\big(F(z)\to P(z)\big)$ says, "for any person, if they are female then they satisfy $P$."   More simply: "Every woman satisfies $P$"

To restrict an existential quantifier, use the conjunctive connective.

$\forall z~\big(M(z)\to Q(z)\big)$ says, "There is a person, they are male and they satisfy $Q$."   More simply: "Some man satisfies $Q$"