Consider the compund propositions given below as: (a) \(p \vee \sim (p \wedge…
2015
Consider the compund propositions given below as:
(a) \(p \vee \sim (p \wedge q)\) (b) \((p \wedge \sim q) \vee \sim (p \wedge q)\)
(c) \(p \wedge (q \vee r)\)
Which of the above propositions are tautologies
Answer: B. Only (a) — Conclusion: Only proposition (a) is a tautology. The provided choice "Only (a)" correctly identifies the tautology. Proposition (a): p ∨ ¬(p ∧ q). Use De…
- A.
(a) and (b)
- B.
Only (a)
- C.
(a) and (b)
- D.
(a), (b) and (c)
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Correct answer: B
Conclusion: Only proposition (a) is a tautology. The provided choice "Only (a)" correctly identifies the tautology.
Proposition (a): p ∨ ¬(p ∧ q). Use De Morgan: ¬(p ∧ q) = ¬p ∨ ¬q, so the expression becomes p ∨ (¬p ∨ ¬q) = (p ∨ ¬p) ∨ ¬q = True ∨ ¬q = True. Therefore (a) is always true (a tautology).
Proposition (b): (p ∧ ¬q) ∨ ¬(p ∧ q). Substitute ¬(p ∧ q) = ¬p ∨ ¬q to get (p ∧ ¬q) ∨ (¬p ∨ ¬q). This simplifies to ¬p ∨ ¬q (absorption). This is not always true; for example, when p = true and q = true the expression is false. Hence (b) is not a tautology.
Proposition (c): p ∧ (q ∨ r) depends on p, q, r. For instance, if p = true, q = false, r = false, then q ∨ r = false and the whole expression is false. Thus (c) is not a tautology.
Final answer: Only proposition (a) is a tautology, so the correct choice is the one that states "Only (a)".
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