Question

If $S^{\ast }(p,q)$ is the dual of the compound statement S (p, q) then $S^{\ast }(\sim p,\sim q)$ is equivalent to

• A. $S(\sim p,\sim q)$
• B. $\sim S(p,q)$
• C. $\sim S^{\ast }(q,q)$
• D. None of these.

Answer 1

The correct option is B $\sim S(p,q)$

Let $S(p,q)\equiv (p\vee \sim q)\wedge \sim p$

$\Rightarrow S(\sim p,\sim q)\equiv (\sim p\vee q)\wedge p$

Now $S^{\ast }(\sim p,\sim q)\equiv (\sim p\wedge q)\vee p$

and $\sim S(p,q)\equiv \sim \left[\right(p\vee \sim q)\wedge \sim p]\equiv \sim (p\vee \sim q)\vee p$

$\equiv (p\wedge q)\vee p$

Hence $S^{\ast }(\sim p,\sim q)\equiv \sim S(p,q)$
Ans: B