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

$\begin{array}{rl}[(P\Rightarrow Q)\Rightarrow R]\vee (\mathrm{\neg }P\vee Q)& \equiv [(\mathrm{\neg }P\vee Q)\Rightarrow R]\vee (\mathrm{\neg }P\vee Q)\\ & \equiv \mathrm{\neg }(\mathrm{\neg }P\vee Q)\vee R\vee (\mathrm{\neg }P)\vee Q\\ & \equiv (P\wedge \mathrm{\neg }Q)\vee R\vee (\mathrm{\neg }P)\vee Q\\ & \equiv [P\vee R\vee (\mathrm{\neg }P)\vee Q]\wedge [(\mathrm{\neg }Q)\vee R\vee (\mathrm{\neg }P)\vee Q]\end{array}$

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

$\begin{array}{rl}A& =P\vee R\vee (\mathrm{\neg }P)\vee Q\\ B& =(\mathrm{\neg }Q)\vee R\vee (\mathrm{\neg }P)\vee Q\end{array}$

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.