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JEE Main202324 Jan 2023Morning ShiftMathematicsMathematical ReasoningActual

The compound statement ( ~ ( P ∧ Q ) ) ∨ ( ( ~ P ) ∧ Q ) ⇒ ( ( ~ P ) ∧ ( ~ Q ) ) is equivalent to

Options

  1. A( ( ~ P ) ∨ Q ) ∧ ( ( ~ Q ) ∨ P )
  2. B( ~ Q ) ∨ P
  3. C( ( ~ P ) ∨ Q ) ∧ ( ~ Q )
  4. D( ~ P ) ∨ Q

Correct answer

A. ( ( ~ P ) ∨ Q ) ∧ ( ( ~ Q ) ∨ P )

Step-by-step solution

We have, P Q ~ P ∧ Q ~ P ∧ Q ~ P ∧ Q ∨ ~ P ∧ Q ~ P ∧ ~ Q ~ P ∧ Q ∨ ~ P ∧ Q ⇒ ~ P ∧ ~ Q T T F F F F T T F T F T F F F T T T T F F F F T F T T T Also, P Q ~ P ~ Q ~ P ∨ Q ~ Q ∨ P ~ P ∧ Q ∧ ~ Q ∨ P T T F F T T T T F F T T F F F T T F F F F F F T T T T T Hence, ~ P ∧ Q ∨ ~ P ∧ Q ⇒ ~ P ∧ ~ Q ≡ ~ P ∧ Q ∧ ~ Q ∨ P .

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