Please explain this statement from Sentential Logic.

Question

On a textbook I am using, there's this question:

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

And the answer is I will not buy the shirt without the pants.

I don't get why with the use of "without"? Why it is not I will not buy the pants or I will buy the shirt with the use of DeMorgan's Law. I know I am missing something, thank you in advance.

Answer 1

Prove that stament is equivalent to $P \rightarrow S.$
Thus, if I buy the pants I will also buy the shirt.
The contrapositive is:
if I do not buy the shirt, then I will not buy the pants.
The given answer indicates that
if I buy the shirt, I will buy the pants.
Apparently it is wrong.

Answer 2

Literally, $\lnot (P \land \lnot S)$ means "It is not the case that I will buy the pants and I will not buy the shirt", which amounts to say "It is not the case that I will buy the pants without the shirt", i.e., in a more concise and elegant way, "I will not buy the pants without buying the shirt".

So, if in the textbook the solution is "I will not buy the shirt without the pants", there is a typo, you have to swap "pants" and "shirt" in the textbook solution.