Science:Math Exam Resources/Courses/MATH220/December 2011/Question 03/Solution 1

Using both hints, we can rewrite the statement into equivalent statements as follow

${\begin{aligned}\left[(P\Rightarrow Q)\Rightarrow R\right]\lor (\lnot P\lor Q)&\equiv [(\lnot P\lor Q)\Rightarrow R]\lor (\lnot P\lor Q)\\&\equiv \lnot (\lnot P\lor Q)\lor R\lor (\lnot P)\lor Q\\&\equiv (P\land \lnot Q)\lor R\lor (\lnot P)\lor Q\\&\equiv [P\lor R\lor (\lnot P)\lor Q]\land [(\lnot Q)\lor R\lor (\lnot P)\lor Q]\end{aligned}}$

The last equivalent statement is of the type A AND B with

${\begin{aligned}A&=P\lor R\lor (\lnot P)\lor Q\\B&=(\lnot Q)\lor R\lor (\lnot P)\lor Q\end{aligned}}$

which both are tautologies since they contain P OR not P and respectively not Q OR Q. So if both A and B are tautologies, then so is the conjunction of them into the statement A AND B.