Need help formalising simple propositional logic sentences
I can understand that the use of 'cannot' is a bit confusing ... it seems to be stronger than just saying that David and Emily are not both happy: they may not both be happy now, sure, but to say that they cannot both be happy seems to say that they can't ever both be happy, i.e. that it is impossible for both to be happy.
In fact, in modal logic you can express these kinds of stronger claims, where:
$\square P$ means "It is necessary that P is true"
$\Diamond P$ means "It is possible that P is true"
Using those symbols, translating "David and Emily cannot both be happy" can be done as:
$\neg \Diamond (r \land p)$
or, equivalently:
$\square \neg (r \land p)$
But, I assume you are currently not doing any model logic at all, since you are just starting with propositional logic. As such, you should really just treat the sentence as "David and Emily are not both happy"
Good for you for noticing that those two sentences are not quite the same thing though!!
The last sentence means that the statement "David is happy and Emily is happy" is false. Thus, it is equivalent to $\overline{p \wedge r}$, or $\bar p \vee \bar r$.
If David and Emily cannot both be happy, then this would be one of the following means of writing it.
$$\neg [p \land r]$$
i.e. the negation of both David and Emily being happy. Since they cannot be happy, then the statement they're both happy is false, and thus we use the negation.
You could hypothetically use each of the possible cases in a sort of list separated by "or" operators. If David and Emily cannot both be happy, then either Emily is happy but David isn't, David is happy but Emily isn't, or neither are. In which case this could be listed as
$$[\neg p \land r] \lor [p \land \neg r] \lor [\neg p \land \neg r]$$
A third way to write it: since Emily or David cannot be both happy, it means at least one is unhappy. That is, either David is unhappy, or Emily is unhappy, or possibly both (but we need not account for this in this way of writing it). Thus another take on this is
$$\neg p \lor \neg r$$
I imagine the first and third would be the "intended" answers for an exercise of this sort since they're the most compact. They also show a nice equality worth keeping in mind:
$$\neg [p \land r] = \neg p \lor \neg r$$
(Footnote: throughout this post, $\neg$ is used as the "negation" operator, since I've seen a few different notations for it.)