First Order Logic parentheses

Question

I have hard time understanding use of parentheses in FOL. Could someone explain to me why in the given example my answer was not correct? What do those parentheses change in the meaning of sentence? EXAMPLE

Answer 1

This question gets an extra layer of frustration, because the possible answers are both missing a parenthesis -- neither of them is a well-formed formula. From context we can infer that the two possible answers should be $$ \forall x (\forall y(\mathrm{Hates}(x, y)) \to \mathrm{Bad}(x)),\\ \forall x (\exists y(\mathrm{Hates}(x, y)) \to \mathrm{Bad}(x)). $$ You selected the former, but the correct answer is the latter. The original sentence was $\forall x \forall y(\mathrm{Hates}(x, y) \to \mathrm{Bad}(x))$. In other words: if you can find any $x$ and any $y$, so that $x$ hates $y$, then $x$ is bad. In particular, if everyone hates Bobby, then everyone is bad, because for $y$ you can always take the value Bobby.

The first of the two options states something different. It says the following: for any person $x$, if they hate everyone, then they are bad. So you could hate Bobby all you want, as long as there is at least one person you don't hate. So you can see that this means something different than the original sentence.

The second sentence states: for any person $x$, if there is someone they hate, then they are bad. This is actually exactly what the first statement tells you: that any $x$ who hates anyone is necessarily a bad person.