Question

If the Boolen expression $(p\Rightarrow q)\iff (q\ast (\sim p\left)\right)$ is a tautology, then the Boolean expression $p\ast (\sim q)$ is equivalent to:

• A. $p\Rightarrow \sim q$
• B. $p\Rightarrow q$
• C. $q\Rightarrow p$
• D. $\sim q\Rightarrow p$

Answer 1

The correct option is C $q\Rightarrow p$

$(p\to q)\iff (q\ast \sim p)$

$p$	$q$	$p\to q$	$q\ast \sim p$	$\sim q$	$\sim q\wedge p$	$\sim (\sim q\wedge p)$
T	T	T	T	F	F	T
T	F	F	F	T	T	F
F	T	T	T	F	F	T
F	F	T	T	T	F	T

$\therefore \sim (\sim q\wedge p)=q\vee \sim p=\sim p\vee q\therefore \ast \text{is equivalent to}\vee \therefore p\ast \sim q=p\vee \sim q$
$=\sim q\vee p$
$=q\Rightarrow p$