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Here are my steps:

(p ∨ q) ∧ (¬p ∨ r) → (q ∨ r)  
¬[ (p ∨ q) ∧ (¬p ∨ r) ] ∨ (q ∨ r)   implication to disjunction  
¬(p ∨ q) ∨ ¬(¬p ∨ r) ∨ (q ∨ r)      demorgans law  
(¬p ∧ ¬q) ∨ (p ∧ ¬r) ∨ (q ∨ r)      demorgans law + double negation

I'm stuck on this last step. The only law that seemed hopeful was the distribution law but that won't even work here. I resorted to using a truth table to prove this but I really want to know if it's possible to shrink this proposition to just true to make it a tautology.

Thank you!

shim
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  • That's weird. Implication to disjunction seems to have failed. – Vons Feb 07 '21 at 07:18
  • Isn't implication to disjunction (and all the other steps) supposed to preserve the truthity of the statement? It turns from F under your conditions to T after applying implication to disjunction. – Vons Feb 07 '21 at 07:27
  • @JMP I looked it up and actually False on the left hand side means that the statement is true. https://philosophy.stackexchange.com/questions/7722/how-would-you-explain-the-implication-disjunction-equivalence/7726 – Vons Feb 07 '21 at 07:37
  • q,r false means the LHS is false and so the statement is true regardless of whether the RHS is true or false. An arrow statement is only false for $T\rightarrow F$. (I can’t explain why. I last did this in like middle school.) – Vons Feb 07 '21 at 08:12

1 Answers1

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You can continue as follows; I think that the steps should all be fairly easy for you to justify.

$$\begin{align*} &(\neg p\land\neg q)\lor(p\land\neg r)\lor(q\lor r)\\ &\big((\neg p\land\neg q)\lor q\big)\lor\big((p\land\neg r)\lor r\big)\\ &\big((\neg p\lor q)\land(\neg q\lor q)\big)\lor\big((p\lor r)\land(\neg r\lor r)\big)\\ &\big((\neg p\lor q)\land\top\big)\lor\big((p\lor r)\land\top\big)\\ &(\neg p\lor q)\lor(p\lor r)\\ &(\neg p\lor p)\lor(q\lor r)\\ &\top\lor(q\lor r)\\ &\top \end{align*}$$

Brian M. Scott
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  • Thank you Brian for the clarification. I guess I did have to use the distribution law but it just didn't click for me. After realizing that by reading your answer, it makes much more sense. – shim Feb 07 '21 at 19:35
  • @shim: You’re welcome; glad it makes more sense now. – Brian M. Scott Feb 07 '21 at 22:17