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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.

  1. Q -> P

ii. Victoria studies hard for the final exam, and does not desperately want to ace it.

  1. P ^ !Q

iii. With Victoria, studying hard for the final exam is necessary but not sufficient for desperately wanting to ace it.

  1. !(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.

Math Student
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1 Answers1

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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.

Rory Daulton
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  • Thanks but How? Thats what I don't understand – Math Student Aug 03 '14 at 20:57
  • is necessary but "not sufficient"(this throws me off) – Math Student Aug 03 '14 at 20:59
  • "P is sufficient for Q" means that if P is true, that is enough to be sure that Q is true. In other words, it means P->Q. "P is necessary and sufficient" means P->Q and Q->P, which can be written P<->Q and said "P is equivalent to Q." Your question's statement "is necessary but not sufficient" means Q->P but not P->Q, or perhaps Q->P but we do not know if P->Q. Is that clear? (A full explanation would take many more words.) – Rory Daulton Aug 04 '14 at 01:03
  • Yes thank, makes sense now! – Math Student Aug 05 '14 at 12:52