The sentences below are either true no matter how P is interpreted or False for some interpretations of P. Decide which one fits into which category.
(a) (∃x P(x)) ⇒ (∀y P(y))
(b) ∀x (P(x) ⇒ ∃y P(y))
(c) ∃x (P(x) ⇒ (∀y P(y))
(d) ∀x∃y (P(x) ⇒ P(y))
(e) ∃y∀x (P(x) ⇒ P(y))
So I thought a) would be false because if P(x) simply exists, it does not necessarily imply that P(y) exists for all y.
For b) I thought it would be true because it is already stated that the sentence applies for all x and therefore it must be true for all interpretations of P.
For c), I thought it was false because I did not think that simply because x existed implied that y existed for all cases.
I thought both d) and e) would be true because as all long as both exist for all y and x, order didn't matter and the implication would simply stand.
However, I am not sure about my reasoning for either and it is very likely I have gotten many wrong.
Any help?
A. Give an example. Try something like P(x) for x is odd.
B. True.
C. True. There are two cases: for all x, P(x) and its negation.
D. True. For each x, let y = x.
E. True. There are two cases: for all x, not P(x) and its negation.
Order of quantifiers makes a big difference.