Question
If $$S\left( {p,q,r} \right) = \left( { \sim p} \right) \vee \left[ { \sim \left( {q \wedge r} \right)} \right]$$ is a compound statement, then $$S\left( { \sim p, \sim q, \sim r} \right)$$ is
A.
$$ \sim S\left( {p,q,r} \right)$$
B.
$$S\left( {p,q,r} \right)$$
C.
$$p \vee \left( {q \wedge r} \right)$$
D.
$$p \vee \left( {q \vee r} \right)$$
Answer :
$$p \vee \left( {q \vee r} \right)$$
Solution :
$$\eqalign{
& S\left( {p,q,r} \right) = \left( { \sim p} \right) \vee \left[ { \sim \left( {q \wedge r} \right)} \right] \cr
& S\left( { \sim p, \sim q, \sim r} \right) = \sim \left( { \sim p} \right) \vee \left[ { \sim \left( { \sim q \, \wedge \sim r} \right)} \right] \cr
& = p \vee \left[ { \sim \left( { \sim q} \right) \vee \sim \left( { \sim r} \right)} \right] = p \vee \left( {q \vee r} \right) \cr} $$