We define two propositions P and Q as follows.
P: Victoria studies hard for the final exam.
Q: Victoria desperately wants to ace the final exam.
(a) Translate each of the following statements into a propositional formula that uses P and Q. No justification is required.
i. Victoria desperately wants to ace the final exam only if she studies hard for it.
- Q -> P
ii. Victoria studies hard for the final exam, and does not desperately want to ace it.
- P ^ !Q
iii. With Victoria, studying hard for the final exam is necessary but not sufficient for desperately wanting to ace it.
- !(P -> Q)
(Im not so sure about this one because it had the wording "necessary but not sufficient")
(b) Choose two statements from part (a) and prove them to be logically equivalent.
I will prove 2 is equivalent to 2.
!P -> Q
<=> !( !P / Q) [-> law]
<=> !!P ^ !Q [De Morgans]
<=> P ^ !Q [Doble Negation]
Hence 2 is equivalent to 3.
For iii, "is necessary but not sufficient" can be reworded as "if." Therefore, the answer to iii is Q->P. You can see this if you see that "is necessary but not sufficient" means "the first must be true for the second to be true" which also means "if the second is true than the first was true" which can be simply said "the second implies the first." Part (b) of the question is then trivial.