I've begun to study Daniel Velleman's "How to Prove It" and I need to clarify a few things and ensure I've gotten the answers correct before I move on.
I've questions about quite a few of the exercises which I will be splitting up over multiple posts, I hope that isnt a problem.
Exercise 1.1
Question 5
Let $P$ stand for the statement "I will buy the pants" and $S$ for the statement "I will buy the shirt". What English sentences are represented by the following expressions?
(a) $\neg (P \wedge \neg S)$
(a) $\neg (P \wedge \neg S)$
The answer in the back of the book says that this statement means:
I won't buy the pants without the shirt
I don't quite understand that, because the statement being negated here is $(P \wedge \neg S)$ and this statement to my understanding means "I will buy the pants but I will not buy the shirt". Thus the statement $\neg (P \wedge \neg S)= \neg P \wedge \neg \neg S$ is a negation of all this and should mean "I will not buy the pants but I will buy the shirt". I assume the double negation before the $S$ will result in a positive statement, in the same way if I said "I will not not eat ice cream", means I will eat ice cream.
Can someone tell me where im going wrong here? Is my answer wrong?
You say that $\neg(P \land \neg S) = \neg P \land \neg\neg S$. This is not true. You have to use de Morgan's law, which will give you $$\neg(P \land \neg S) = \neg P \lor \neg\neg S = \neg P \lor S = P \rightarrow S,$$ meaning "If I buy the pants, then I also buy the shirt", or equivalently "I won't buy the pants without also buying the shirt".
A more intuitive way to think about it:
$\neg (P \land \neg S)$ means "it is not the case that both $P$ and not-$S$ are true", hence we must have that "either $P$ is false or not-$S$ is false", which is the same as $\neg P \lor \neg\neg S$. So far, this is just de Morgan's law. $\neg\neg S$ is, as you said, the same as $S$, so we get $\neg P \lor S$. This means "either I don't buy the pants, or I buy the shirt", which in particular means that if I do buy the paints, then I will also buy the shirt. This is the same as saying "I won't buy the pants without the shirt".