# How to simplify complex logic connectives? (DNF)

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

• 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. – mathreadler Feb 28 '18 at 6:07

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.