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GATE1993-8.2

17 votes
2.8k views

The proposition $p \wedge (\sim p \vee q)$ is:

  1. a tautology

  2. logically equivalent to $p \wedge q$

  3. logically equivalent to $p \vee q$

  4. a contradiction

  5. none of the above

in Mathematical Logic
edited by 2.8k views
0
why not option a? it is modus ponens clearly and hence it is a tautology... pls pls clear my doubt...
0
modus ponens is an inference rule to conclude something.. It is not used to show tautology. Here, put 'P'  as False. It is not a tautology.

4 Answers

18 votes
 
Best answer

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

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

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

$\equiv (p \wedge q)$


Hence, Option(B) logically equivalent to $ (p \wedge q)$.


edited by
0
why not option a? it is modus ponens clearly and hence it is a tautology... pls pls clear my doubt...
0
Just substitute values debasree , it’s not ALWAYS TRUE, so not tautology.
13 votes
OPTION (B)
11 votes
p ^ (~p v q)

= (p ^ ~p) v (p ^ q)

= False V (p^q)

= (p^q)
1 vote
$P \wedge (\neg P \vee Q ) \equiv  ( P \wedge \neg P ) \vee ( P \wedge Q )$

                         $\equiv F \vee ( P \wedge Q )\:\:\: \bigg[\because (P \wedge \neg P) \: \text{ is always False and} \: (P \vee \neg P)\:\text{ is always True} \bigg]$

                        $\equiv ( P \wedge Q )$

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