Question

Which one of the following is a tautology ?

• A. $(P ∧ (P \to Q)) \to Q$
• B. $P ∨ (P ∧ Q)$
• C. $Q \to (P ∧ (P \to Q))$
• D. $P ∧ (P ∨ Q)$

Answer 1

Answer: (A)

$(P ∧ (P \to Q)) \to Q$

Answer 2

$\left(1\right)\,P\,\wedge \left(P\,\vee\,Q\right)=P$
$\left(2\right)\,P\,\vee \left(P\,\wedge\,Q\right)\equiv P$
$\left(3\right)\,Q \rightarrow \left(P\,\wedge\left(P \rightarrow Q\right)\right)$
$\equiv Q \rightarrow \left(P\,\wedge\left(\sim P\,\vee\,Q\right)\right)\equiv Q \rightarrow \left(P\,\wedge\,Q\right)$
$\equiv\left(\sim Q\right)\vee\left(P\,\wedge\,Q\right)\equiv\left(P\,\vee\left(\sim Q\right)\right)$
$\left(4\right)\,\left(P\,\wedge \left(P \rightarrow Q\right)\right) \rightarrow Q$
$\equiv\left(P\,\wedge \left(\sim P\,\vee\,Q\right)\right) \rightarrow Q\equiv \left(P\,\wedge\,Q \right) \rightarrow Q$
$\equiv\left(\left(\sim P\right)\vee\left(\sim Q\right)\right)\vee\,Q\equiv\left(\sim P\right)\vee\,t\equiv t$

So, the correct option is (A): $(P ∧ (P \to Q)) \to Q$