Q.3 Using truth table, prove that the statement patterns \(p \leftrightarrow q\) and \((p \wedge q) \vee (\sim p \wedge \sim q)\) are logically equivalent.
Best answer
Answer: Columns 3 and 8 of the truth table are identical, so the two statement patterns are logically equivalent.
| \(p\) | \(q\) | \(p \leftrightarrow q\) | \(p \wedge q\) | \(\sim p\) | \(\sim q\) | \(\sim p \wedge \sim q\) | \((p \wedge q) \vee (\sim p \wedge \sim q)\) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | F | F | F | T |
| T | F | F | F | F | T | F | F |
| F | T | F | F | T | F | F | F |
| F | F | T | F | T | T | T | T |
All entries in columns 3 and 8 match, hence \(p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)\).
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