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The negation of the statement $$(p \vee q) \wedge (q \vee ( \sim r))$$ is :

JEE · Math · previous-year question

  1. A.$$(( \sim p) \vee r)) \wedge ( \sim q)$$correct
  2. B.$$(p \vee r) \wedge ( \sim q)$$
  3. C.$$(( \sim p) \vee ( \sim q)) \vee ( \sim r)$$
  4. D.$$(( \sim p) \vee ( \sim q)) \wedge ( \sim r)$$

Answer

A. $$(( \sim p) \vee r)) \wedge ( \sim q)$$

Explanation

The negation of the statement $(p \vee q) \wedge(q \vee(\sim r))$ is $$ \begin{aligned} & =\sim[(p \vee q) \wedge(q \vee(\sim r))] \\\\ & =\sim[(p \vee q) \vee \sim(q \vee(\sim r)] \\\\ & =((\sim p) \wedge \sim q)) \vee((\sim q) \wedge r)) \end{aligned} $$ Apply distribution law, $$ \begin{aligned} & =\sim q \wedge((\sim p) \vee r)) \\\\ & =((\sim p) \vee r) \wedge(\sim q) \end{aligned} $$

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