Equivalence rule question: Use the equivalence rules to push all occurrences of the negation symbol ¬ next to the atoms in the formulas below

Question

3. [MEDIUM, 8 mins] Use the equivalence rules to push all occurrences of the negation symbol ¬ next to the atoms in the formulas below: • ¬((P → Q) ∨ ((P → R) ∧ ¬P))

Answer 1

Do you mean this sort of reduction?

$\neg((P \rightarrow Q) \vee ((P \rightarrow R) \wedge \neg P))$

$\neg(P \rightarrow Q) \wedge \neg ((P \rightarrow R) \wedge \neg P))$

$\neg(P \rightarrow Q) \wedge (\neg (P \rightarrow R) \vee \neg \neg P))$

$\neg(P \rightarrow Q) \wedge (\neg (P \rightarrow R) \vee P))$

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