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?!?!*
Thanks.
Best Answer
As a general rule of thumb, most (negated) universal English language sentences will be translated as a (negated) universal conditional, while (negated) existential sentences will be translated as a (negated) existential conjunction. If you get universal conjunctions or existential conditionals in your translation, then you might want to double check your answer. For instance, the universal fragment of your first answer, "$\forall y (D(y) \wedge S(x,y))$," translates as "Everyone is in the discrete math class and $x$ is smarter than everyone," whereas you wanted to say "$x$ is smarter than anyone who is in the discrete math class," or equivalently, "Everyone who is in the discrete math class is such that $x$ is smarter than them."
In general, if your English sentence has the form "Everything that is $\phi$ is $\psi$," your translation will be along the lines "$\forall x (\phi \rightarrow \psi)$," whereas "$\forall x (\phi \wedge \psi)$" says, "Everything is both $\phi$ and $\psi$," which is a much stronger claim. Similarly, "Some $\phi$ is $\psi$" gets translated as "$\exists x (\phi \wedge \psi)$," whereas "$\exists x (\phi \rightarrow \psi)$" says, "Something is such that if it's $\phi$, then it's $\psi$," or rather "Something is either not $\phi$ or $\psi$," which is a much weaker claim. "No one that is $\phi$ is $\psi$" is typically translated as "$\forall x (\phi \rightarrow \neg\psi)$". But there are equivalent formulations of each of these sentences as well.
Your edited answer is almost correct. However, "$S(y,x)$" is not logically equivalent to "$\neg S(x,y)$", and you want the effect that your $x$ is only not smarter than your $y$. But if you just replace "$S(y,x)$" with "$\neg S(x,y)$", then your new answer will be correct. Also acceptable translations of this sentence include: