Consider the following statements.
(i) I buy all interesting movies.
(ii) I have watched all movies by Wolfgang Petersen.
(iii) For any movie, either I buy it (i.e., buy its DVD) or watch it.
(iv) All interesting movies have a rating of more than 8 on imdb.com.
(v) The movie Troy has a rating of 7.5 on imdb.com.
Let M be the set of all movies. Let the boolean function I(m) return true if m is interesting. Similarly, consider other boolean functions B(m), W (m), P (m), E(m) to denote me having bought the movie, me having watched the movie, the movie made by W. Petersen, and the movie having an 8 plus rating.
(a) Write each of the above five statement in pure logical form using appropriate quantifications.
i) ForAll m in M, B(m) —> I(m)
ii) ForAll m in M, P(m) —> W(m)
iii) ForAll m in M, B(m) OR W(m)
iv) ForAll m in M, I(m) —> E(m)
v) ThereExists a movie m in M (named Troy), such that ~E(m)
(b) Can we conclude the statement “Troy is by Wolfgang Petersen”? [Hint: If p → q and r → s then (p ∪ r) ! (q ∪ s).] If yes, show the sequence of inferences. If not, justify your answer.
Since all interesting movies have a rating of more than 8 on imdb.com, this means that there can be other movies that have a rating higher than 8 that are not interesting, and also means that all movies that have a rating of less than or equal to 8 are not interesting. Troy has a rating of 7.5 on imdb.com, which is less than 8, and therefore is not interesting. From (i) and (iii), Since Troy is not interesting, I did not buy it, and therefore must have watched it. Just because I have watched all movies by Wolfgang Petersen, that does not mean that I have never watched a movie that is not by him, so since I watched Troy, it could very well not be by Wolfgang Peterson. There is not enough information to conclude that Troy is by Wolfgang Petersen.
i) should be $\forall m \in M (I(m) \rightarrow B(m))$
for v), you should probably use a constant to denote the specific Troy movie .. so let's use $t$. Then we get: $\neg E(t)$ ('Troy did not get a rating above 8 on imdb)
Your explanation for b) needs to change a bit since you can't conclude that you don't buy any uninteresting movies.