Write the English language statement as a predicate wff (the domain of interpretation is the whole world).
All games are more fun than some movie.
G(x): x is a game
M(x): x is a movie
F(x,y): x is more fun than y
My attempt: $$(\forall x)(\exists y)\Bigl(M(y)\land \bigl(G(x)\rightarrow F(x,y)\bigr)\Bigr)$$ The textbook's answer: $$(\forall x)(\exists y)\Bigl(G(x)\rightarrow\bigl(M(y)\land F(x,y)\bigr)\Bigr)$$ I was able to prove with propositional expressions $\bigl[M\land(G\rightarrow F)\bigr] \rightarrow \bigl[G\rightarrow(M\land F)\bigr]$, but not the other way around. I made a truth table, and the only differences between the two are: when $M$ and $G$ are both false, the book's expression takes on $True$ "truth values" and my expression has $False$ truth values.
To me, it seems like my expression is more "specific" in that its truth involves a smaller set of possible interpretations. The truth of my expression requires $y$ to be a movie, whereas according to the book, the expression should be true even if $y$ is not a movie, and $x$ is not a game.
Where did I go wrong? Or more specifically, what steps could I have taken when evaluating my wff that would have helped me realize I made a mistake?
The difference between your answer and the book's answer is very slight, but consider a situation where there are no games, and no movies, but where the domain isn't completely empty either (which is what logic typically assumes anyway). In that case, the English sentence expresses something that is (trivially) true: All (zero) games are indeed more fun than some movie. However, your sentence would be false, since your sentence claims that for any object from the domain that I pick (and again, there is at least one of those), there needs to be a movie. But since there aren't any movies, your sentence would evaluate false in such a domain. So: your sentence does not capture the English sentence.
Instead, you really only want to be looking for movies for those objects that are games. So, first you should make sure that $x$ is a game, and if so, then you should be able to locate a movie for it. This is exactly what the book's answer does.