Translating the 2 following sentences in propositional calculus

Question

what do the two following sentences look like if you use logical connectors?

1. q is necessary but not sufficient for p.

2. q is sufficient but not necessary for p.

I know how to do q is necessary for p, and q is sufficient for p, as well as q is necessary and sufficient for p, but I'm getting lost when you throw a not in there.

Answer 1

OK, so you know that:

'q is sufficient for p' translates as: $q \rightarrow p$

and

'q is necessary for p' translates as: $p \rightarrow q$

So:

1. 'q is necessary but not sufficient for p' translates as: $(p \rightarrow q) \land \neg (q \rightarrow p)$

2. 'q is sufficient but not necessary for p.' translates as: $(q \rightarrow p) \land \neg (p \rightarrow q)$

Because a 'but' in English, translates as a simple $\land$ in logic. E.g. if I say: 'I am rich but not happy', I mean $R \land \neg H$ (with $R$ for me being rich, and $H$ for me being happy of course)