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I am having trouble finding which of the following statements are true:

Which of the following statements are true?

  • [a] Pizza does NOT have mushrooms
  • [b] Pizza does have mushrooms AND bacon
  • [c] Pizza does have mushrooms OR bacon
  • [d] Pizza does have mushrooms AND NOT shrimps
  • [e] Pizza does NOT have mushrooms OR NOT bacon

I've tried to solve this problem like below.

  1. I assigned a letter to each unique statement:
    • P=Pizza does have mushrooms
    • Q=Pizza does have bacon
    • R=Pizza does have shrimps
  2. I've translated each statement in terms of P,Q,R like this: $$a=\neg P$$ $$b=P \wedge Q$$ $$c=P \vee Q$$ $$d=P \wedge \neg R$$ $$e=\neg P \vee \neg Q$$

Because I don't know the value of truth of neither P,Q or R I could just assume "what if P is true" and then "what if Q is true" and then "what if R is true" and all other combinations (which is 2^3 because |{T,F}|=2 and |{P,Q,R}|=3).

Just having that table it doesn't (seem to) help me deciding which are really true because in one case or another any of them can be true, depending on our assumption regarding the value of truth for P,Q,R.

According to the book's key the answer should be [c] and [e]. I can remark in my table that if I have to compare/count which of [a]->[e] have more T than others then [c] and [e] are the winners. But this is not a formal proof, does it?

Please give me a tip or point me into the right direction. Thx.

Mauro ALLEGRANZA
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Eugen Mihailescu
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  • I think you should rephrase the question, as any statement by itself may be true of false. Perhaps you meant to ask something like: "What is the largest number of statements that can be combined into a single valid statement, and what are those statements?" (in which case, the answer is $3$ statements - either statements #$2,3,4$ or statements #$3,4,5$). – barak manos Jun 12 '14 at 09:34
  • I have quoted the problem exactly as is written in the book. I don't have any other information. I assumed that maybe I don't see the answer but now, because you too have the same problem like I had, I am starting to think that the exercise in the book was badly formulated. – Eugen Mihailescu Jun 12 '14 at 09:37

1 Answers1

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What we can say is :

if [a] is true, then only [c] and [e] can be true.

According to your (correct) formulation, if we assume $\lnot P$, then both $P \land Q$ amd $P \land \lnot R$ are false, because a conjunction is true only when both consjuncts are.

Thus, we are left with [e] : $\lnot P \lor \lnot Q$, that is true due to [a] : $\lnot P$.

For [c] : $P \lor Q$, it can be true if $Q$ is.

Mauro ALLEGRANZA
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