I am self-studying Daniel Velleman's "How to Prove It."
In the exercises for section 2.1, for question # 1b, I got a different answer than he did (his answer is in the back of the book).
I think my answer is equivalent to his, and I also think I see yet another equivalent answer.
But since I'm learning, I'm not confident enough that my answers really are equivalent, so I hope someone here can help me.
The question is to "analyze the logical form" of the following statement: "Nobody in the calculus class is smarter than everybody in the discrete math class."
Velleman's answer is: $$ \lnot \exists x (C(x) \land \forall y (D(y) \to S(x,y)))) $$
I can see that, but it appears the following are fine too. In fact, it looks to me like you have a choice whether or not to use If-Then at all, and if you want to use it, you have 2 separate places where you could put it.
Is this also correct? $$ \lnot \exists x (C(x) \land (\forall y (D(y) \land S(x,y)))) $$
And is this also correct? $$ \forall x (C(x) \to (\lnot \forall y D(y) \land S(x,y))) $$
EDIT: Here's a third attempt, added later. Is this equivalent? $$ \forall x (C(x) \to \exists y (D(y) \land S(y, x))) $$ where S(y,x) is defined as "y is as smart as or smarter than x"
If those are not correct, please help me understand why not?!?!*