Question

Match the LIST-I with LIST-II. 

LIST-I	LIST-II
A. \(p \cdot q\)	I. \(\sim(\sim p \cdot \sim q)\)
B. \(p \vee q\)	II. \((p \cdot q) \vee (\sim p \cdot \sim q)\)
C. \(p \supset q\)	III. \(\sim(\sim p \vee \sim q)\)
D. \(p \equiv q\)	IV. \(\sim p \vee q\)

Choose the correct answer from the options given below.

• A. A-IV, B-II, C-I, D-III
• B. A-III, B-I, C-IV, D-II
• C. A-I, B-III, C-II, D-IV
• D. A-IV, B-III, C-I, D-II

Hint:

Remember: \[ p \supset q \equiv \sim p \vee q \] and \[ p \equiv q \equiv (p \cdot q)\vee(\sim p \cdot \sim q) \]

Answer 1

The Correct Option is B

Concept:
In symbolic logic, propositions can be represented by logically equivalent forms. Important equivalences include De Morgan's laws and definitions of implication and equivalence.

Step 1: Match conjunction.
\[ p \cdot q \] By De Morgan's law: \[ p \cdot q \equiv \sim(\sim p \vee \sim q) \] So: \[ A \rightarrow III \]

Step 2: Match disjunction.
\[ p \vee q \] By De Morgan's law: \[ p \vee q \equiv \sim(\sim p \cdot \sim q) \] So: \[ B \rightarrow I \]

Step 3: Match implication.
\[ p \supset q \] This is equivalent to: \[ \sim p \vee q \] So: \[ C \rightarrow IV \]

Step 4: Match equivalence.
\[ p \equiv q \] This means both are true together or both are false together: \[ (p \cdot q) \vee (\sim p \cdot \sim q) \] So: \[ D \rightarrow II \]

Step 5: Final matching.
The correct matching is: \[ A-III,\ B-I,\ C-IV,\ D-II \] Hence: \[ \boxed{\text{(B)}} \]