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For this logic connective sentence:

  1. $(P \wedge Q \wedge R) \vee (P \wedge Q \wedge \neg R) \vee (\neg P \wedge Q \wedge R) \vee (\neg P \wedge \neg Q \wedge R)$

How can I simplify the above (or find its equivalent) to:

  1. $(P \wedge Q) \vee (\neg P \wedge R)$

I tried to expand it to disjunction normal form....but it will take a very long time to expand the entire sentence 1. Can anyone please suggest a method that allows the simplification of sentence 1 to sentence 2?

Thanks

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    $\begingroup$ I have not studied logic so much but with boolean algebra you can simplify $ac+ac^{-1}=a(c+c^{-1})=a$. I think it is engineering version of the same thing. $\endgroup$ – mathreadler Feb 28 '18 at 6:07
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As Mathreadler said, we can use distributive property to get: $$(P\land Q \land (R \lor \lnot R)) \lor (\lnot P \land R \land (Q\lor \lnot Q))$$

And since $A \lor \lnot A $ is a tautology, we get the required answer.

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