GATE1993-8.2

Question

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

• A. a tautology

• B. logically equivalent to $p \wedge q$

• C. logically equivalent to $p \vee q$

• D. a contradiction

• E. none of the above

Comments

why not option a? it is modus ponens clearly and hence it is a tautology... pls pls clear my doubt...

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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.

Answer 1

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

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

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

$\equiv (p \wedge q)$


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

Comments

why not option a? it is modus ponens clearly and hence it is a tautology... pls pls clear my doubt...

Answer 2

OPTION (B)

Answer 3

p ^ (~p v q)

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

= False V (p^q)

= (p^q)

Answer 4

$P \wedge (\neg P \vee Q ) \equiv  ( P \wedge \neg P ) \vee ( P \wedge Q )$

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

                        $\equiv ( P \wedge Q )$