Sentential Logic Below is a question comes from the book How to Prove It written by Daniel J. Velleman.

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?
$\lnot(P \land \lnot S)$

The given answer is, “I won't buy the pants without the shirt.“
I don't understand the answer. Can someone explain? Or maybe the answer is wrong?
 A: Let's work from the inside out:


*

*$S$ means: "I will buy the shirt."

*$\neg S$ means: "I won't buy the shirt."

*$P \land \neg S$ means: "I will buy the pants and I won't buy the shirt." Another way to say this is: "I will buy the pants but not the shirt." (or, equivalently, "I will buy the pants without the shirt.").

*$\neg(P \land \neg S)$ means: "I won't buy the pants without the shirt."


Due to the ambiguity of the English language, there will certainly be multiple "translations" and no unique "answer". Later on, you might learn that:
$$
\neg(P \land \neg S) \quad \equiv \quad \neg P \lor S \quad \equiv \quad P \rightarrow S
$$
Thus, a (perhaps more intuitive) valid translation of $\neg(P \land \neg S)$ would be: "If I buy the pants, then I must also buy the shirt."
A: Remember that, by DeMorgan's Laws,
$$\lnot (A\land B)\iff(\lnot A\lor\lnot B),$$
so
$$\lnot (P\land \lnot S)\iff \lnot P\lor(\lnot\lnot S)\iff\lnot P\lor S.$$
Our goal is to describe what it means for $\lnot (P\land \lnot S)$ to be true. So, let's agree that it is true, and see what we conclude.
If you don't buy the pants, then $\lnot P$ is true, and hence so is $\lnot P\lor S$, regardless of what $S$ is. However, if you do buy the pants, then $\lnot P$ is false, and in this case, the only way $\lnot P \lor S$ can be true is if $S$ is true, i.e. if you buy the shirt. 
Thus, to say that $\lnot (P\land \lnot S)$ is true is to say that maybe you won't buy the pants, maybe you will; but if you do, you'll definitely also buy the shirt.
