(1) $(\sim p)\land(p\lor q)\to q$
(2) $(q\to p)\ \lor\sim(p\to q)$
(3) $(\sim q)\lor(p\land q)\to q$
(4) $(p\to q)\land(q\to p)$
Ans. (1) $(\sim p)\land(p\lor q)\to q$
Sol. (i) $\sim p\land(p\lor q)\to q$
$(\sim p\land p)\lor(\sim p\land q)\to q$
$C\lor(\sim p\land q)\to q$
$(\sim p\land q)\to q$
$\sim(\sim p\land q)\lor q$
$=(p\lor\sim q)\lor q=p\lor t=t$
(iii) $(\sim q)\lor(p\land q)\to q$
use $\sim(p\to q)=p\land\sim q\implies p\to q = \sim p\lor q$
$=(\sim q\lor p)\land(\sim q\lor q)\to q$
$=(\sim q\lor p)\to q$
$=(q\land\sim p)\to q$
$=q$
(ii) and (iv)
p | q | $p\to q$ | $q\to p$ | $\sim(p\to q)$ | $(p\to q)\land(q\to p)$ | $(q\to p)\ \lor\sim(p\to q)$ |
T | T | T | T | F | T | T |
T | F | F | T | T | F | T |
F | T | T | F | F | F | F |
F | F | T | T | T | T | T |