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. Thank you.
Question 6
Let S stand for the statement "Steve is happy" and G for "George is happy". What English sentences are represented by the following expressions:
(a) $(S\vee G) \wedge (\neg S \vee \neg G)$
(b) $[S \vee (G \wedge \neg S)] \vee \neg G$
(c) $S \vee [G \wedge (\neg S \vee \neg G)]$
(a) $(S\vee G) \wedge (\neg S \vee \neg G)$
This translates to:
Either Steve or George are happy and Steve or George are not happy. This seems very ill-worded to me, is there a better way to phrase this?
(b) $[S \vee (G \wedge \neg S)] \vee \neg G$
This translates to:
Either Steve is happy or George is happy and Steve is not happy, OR George is not Happy.
(c) $S \vee [G \wedge (\neg S \vee \neg G)]$
This translates to:
Either Steve is happy OR George is happy and Steve is not happy or George is not happy.
Again I don't really understand how to phrase this, the nested statements make things difficult for me, if someone could elucidate this for me it would be really appreciated
a) at least one of the two is happy and at least one of the two is not, therefore one is happy and the other is not.