Hi there I need some help verifying my answers to two compound propositions. I need to use p, q, r, and s to represent the following two sentences.
P denotes "you passed cs2233" q denotes "you passed cs3333" r denotes "you can register for cs3343" s denotes "you understand Boolean algebra"
(1) You can't register for cs3343 only if you haven't passed both cs2233 and cs3333. I'm unsure if the statement should be read as ¬r→(¬p∧¬q) or (¬p∧¬q)→¬r. Should the instance of "only if" mean "if r then ..." or should it be interpreted as "if ... then r".
(2) if you can register for cs3343 then you have passed cs2233 and you understand Boolean algebra if you passed cs2233. I'm confident that this one should be (r→p)∧(p→s), but would like to be sure.
For (1), a statement of the form "$X$ only if $Y$" can be interpreted as "$X$ implies $Y$." Therefore, we can write $\neg r \to (\neg p \wedge \neg q)$. There are, of course, quite a few equivalent ways of stating that.
Your answer for (2) looks correct to me.