Question

Let \(p\) and \(q\) be two propositions. Consider the following two formulae in propositional logic.
\[ S_1 : (\neg p \wedge (p \vee q)) \rightarrow q \] \[ S_2 : q \rightarrow (\neg p \wedge (p \vee q)) \] Which one of the following choices is correct?

• A. Both \(S_1\) and \(S_2\) are tautologies.
• B. \(S_1\) is a tautology but \(S_2\) is not a tautology.
• C. \(S_1\) is not a tautology but \(S_2\) is a tautology.
• D. Neither \(S_1\) nor \(S_2\) is a tautology.

Hint:

An implication is a tautology if it is true for all possible truth values of its variables.

Answer 1

The Correct Option is B

Step 1: Simplify statement S₁.
Consider the antecedent of S₁:

¬p ∧ (p ∨ q)

Using the distributive law:

(¬p ∧ p) ∨ (¬p ∧ q)

Since (¬p ∧ p) is always false, the expression simplifies to:

¬p ∧ q

Hence,

S₁ : (¬p ∧ q) → q

This implication is always true because whenever (¬p ∧ q) holds, q must be true. Therefore, S₁ is a tautology.

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Step 2: Analyze statement S₂.

S₂ : q → (¬p ∧ (p ∨ q))

Consider the valuation: p = true, q = true.

Then:

¬p = false,   (p ∨ q) = true

So:

¬p ∧ (p ∨ q) = false

Hence, the implication becomes:

true → false, which is false.

Therefore, S₂ is not a tautology.

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Final Conclusion:
S₁ is a tautology, but S₂ is not a tautology.

Final Answer: (B)