Question

P and Q are two propositions. which of the following logical expressions are equivalent?
1. $P\vee \sim Q$
2. $\sim (\sim P\wedge Q)$
3. $(P\wedge Q)\vee (P\wedge \sim Q)\vee (\sim P\wedge \sim Q)$
4. $(P\wedge Q)\vee (P\wedge \sim Q)\vee (\sim P\wedge Q)$

• A. Only 1 and 2
• B. Only 1, 2 and 3
• C. Only 1, 2 and 4
• D. All of these

Answer 1

The correct option is B Only 1, 2 and 3

1. $P\vee \sim Q\equiv P+Q^{\prime }$
2. $\sim (\sim P\wedge Q)\equiv (P^{\prime }Q{)}^{\prime }\equiv P+Q^{\prime }$
3. $(P\wedge Q)\vee (P\wedge \sim Q)\vee (\sim P\wedge \sim Q)$
$\equiv PQ+P{Q}^{\prime }+P^{\prime }{Q}^{\prime }$
$\equiv P(Q+Q^{\prime })+P^{\prime }{Q}^{\prime }$
$\equiv P+P^{\prime }{Q}^{\prime }$
$\equiv (P+P^{\prime })(P+Q^{\prime })$
$\equiv p+Q^{\prime }$
4. $(P\wedge Q)\vee (P\wedge \sim Q)\vee (\sim P\wedge Q)$
$\equiv PQ+P{Q}^{\prime }+P^{\prime }Q$
$\equiv P(Q+Q^{\prime })+P^{\prime }Q$
$\equiv P+P^{\prime }Q$
$\equiv (P+P^{\prime })(P+Q)=P+Q$

Clearly (i), (ii) and (iii) are equivalent. Correct choice is (b).