Question

If $S ^{*}(p, q, r)$ is the dual of the compound statement $S ( p , q , r )$ and $S (p, q, r)=\sim p \wedge[\sim(q \vee r)]$ then $S ^{*}(\sim p, \sim q, \sim r)$ is equivalent to -

• A. $S (p, q, r)$
• B. $\sim S (\sim p, \sim q, \sim r)$
• C. $\sim S (p, q, r)$
• D. $S ^{*}(p, q, r)$

Answer 1

Answer: (C)

$S ^{*}(p, q, r)=\sim p \vee[\sim(q \wedge r)]$
$S ^{*}(\sim p, \sim q, \sim r)=\sim(\sim p) \vee[\sim(\sim q \wedge \sim r)]$
$=p \vee[q \vee r]$
$\sim S (p, q, r)=\sim[\sim p \wedge[\sim(q \vee r)]]$
$=\sim(\sim p) \vee \sim[\sim(q \vee r)]=p \vee(q \vee r)$
$= S ^{*}(\sim p, \sim q, \sim r)$