So I just started reading How to Prove It and I'm kind of stuck already on chapter 1 :(. The exercise question is 4.
Analyze the logical forms of the following statements:
a) Either both
Ralph and Ed are tall, or both are handsome
b) Both Ralph and Ed are
either tall or handsome
c) both Ralph and Ed are neither tall nor
handsome
d) Neither Ralph nor Ed is both tall and handsome.
For a) I thought it was P ∨ Q, with P being Ralph and Ed are tall and Q is Ralph and Ed are handsome.
For c) I wrote ¬(P∧Q) with P and Q being the same as a).
My first question is are my two answers correct?
Also the solution for b) and d) is that they aren't well formed. Why is this? (I feel like b is basically the same as a and d is basically the same as c)?
Best Answer
b) is not the same as a): If Ralph is tall but not handsome, and Ed is handsome but not tall, then a) is false (it is not true that both are tall, and it is also not true that both are tall) but b) is true it is true that Ralph is either tall or handsome, and same goes for Ed)
Also, c) is not $\neg(P \land Q)$: you can use the same counterexample: Suppose Ralph is tall but not handsome, and Ed is handsome but not tall. Then $P$ and $Q$ are both false, and hence $P \land Q$ is false as well, and that makes $\neg(P \land Q)$ true. But, in this situation, clearly c) is false: c) is saying that Ralph is neither tall nor handsome, and same for Ed, but in this scenario that is not the case.
Finally, note that this very same scenario also shows that c) and d) are not the same. In this scenario, c) is false, but d) is true.
The key to solving all this is as Mauro indicated: you really need four atomic statements: one for Ralph being tall, one for Ralph being handsome, and same for Ed. With that, you can carefully distinguish between all these 4 different statements.
Good luck!!