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So I am trying to solve this problem and not quite able to figure out the answer. So I'm not sure what two statements I could construct in my truth table to prove these true or false. I believe that $Q$ is not equal to $R$ because OR statements are inclusive. And I believe if we used the AND operators they would be equal. The statement I was thinking about constructing to prove this was if $R \land (P \lor Q) / Q \land (P \lor R)$? Any help would be appreciated

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    $\begingroup$ Doesn't follow, whether you use And or Or. For the Or case, consider the truth-assignment $P \equiv R \equiv \top$ and $Q \equiv \bot$. For the second, consider $P \equiv Q \equiv \bot$ and $R \equiv \top$. $\endgroup$ May 6 '14 at 8:42
  • $\begingroup$ @HunanRostomyan You could post that as an answer. $\endgroup$ May 6 '14 at 14:08
  • $\begingroup$ Could we say that if we assume Q and R to both be true that when we put them into the "AND" statements it means that P would always have to be true therefore they are not equivalent? Then for the "OR" statements if we assume both Q and R are false then if (P OR Q) is going to be equal to (P OR R) P would always have to be false which we know is also not the case therefore both statements show that Q != R $\endgroup$
    – Michael
    May 6 '14 at 15:55
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If P is true, both P OR Q and P OR R are both true regardless of what Q and R may be. Similarly for the AND case, both are false if P is false.

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  • $\begingroup$ @DougSpoonwood How does this contradict what I said? I showed cases where both can be equal without $Q=R$. Which happens to be what the question is about. $\endgroup$
    – Mike
    May 6 '14 at 16:20
  • $\begingroup$ I think I misread your answer. Sorry about that, and for down-voting it. $\endgroup$ May 6 '14 at 16:44
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AND:

"The earth is round" AND "I am white" = FALSE = "The earth is round" AND "I am black"

OR:

"The moon is made of cheese" OR "I am white" = TRUE = "The moon is made of cheese" OR "I am black"

In both cases, Q and R are not the same (they are "I am white" and "I am black", respectively).

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