Negation of the Boolean expression p ⇔ (q ⇒ p) is

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  1. (∼p) ∧ q
  2. (p) ∧ (∼q) 
  3. (∼p) ∨ (∼q)
  4. (∼p) ∧ (∼q)

Answer (Detailed Solution Below)

Option 4 : (∼p) ∧ (∼q)
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Detailed Solution

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Calculation

\(\sim (p \leftrightarrow (q \rightarrow p))\)

⇒ \(\sim (p \leftrightarrow q) = (p \land \sim q) \lor (q \land \sim p)\)

⇒ \(\sim (p \leftrightarrow (q \rightarrow p)) = (p \land \sim (q \rightarrow p)) \lor ((q \rightarrow p) \land \sim p)\)

⇒ \((p \land \sim (q \rightarrow p)) = p \land (q \land \sim p) = (p \land \sim p) \land q = c\)

⇒ \((q \rightarrow p) \land \sim p = (\sim q \lor p) \land \sim p = \sim q \land \sim p = \sim p \land \sim q\)

\(= (\sim p \land \sim q) \lor (\sim p \land p) = \sim p \land \sim q\)

⇒ \(\sim (p \leftrightarrow (q \rightarrow p)) = c \lor (\sim p \land \sim q) = \sim p \land \sim q\)

Hence option 4 is correct

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