I need to express the following into predicate logic: "For every movie, there are at least two different students who have seen it."
The following predicates are given and assumed:
- D(x) is true exactly when x is a student
- M(x) is true exactly when x is a movie
- S(x,y) is true exactly when x has seen y
I've done some "studying" and research and came to a couple conclusions. First, it seems that this problem will require is two Existential Quantifiers for the two different students and a single Universal Quantifier for every movie.
My first impressions were to structure it this way:
- ∃x∃y : There exists a student x and a student y
- ∃x∃y(∀xM(x) ∧ D(x) ∧ D(y) → S(x,y) x ≠ y) : There exists a student x and a student y where for all movies and a student D(x) and student D(y) have seen it where and x is not equal to y.
However, when reading it through the S(x,y) is confusing as in assumed predicates, x has seen y. So, in the logic I wrote above it's read as "student x has seen student y" I think. I'm not sure if I'm on the right track or if I'm completely wrong. Any help or clarification is greatly appreciated.