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Which of the following is a tautology ?

(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) $(p\to q)\land(q\to p)$
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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)

pq$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)$
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