Consider \(\alpha, \beta, \gamma\) as logical variables. Identify which of the…
2022
Consider \(\alpha, \beta, \gamma\) as logical variables. Identify which of the following represents correct logical equivalence :
(A) \((\alpha \wedge(\beta \vee \gamma)) \equiv((\alpha \wedge \beta) \vee(\alpha \wedge \gamma))\)
(B) \((\alpha \vee \beta) \equiv \neg \alpha \vee \beta\)
(C) \((\alpha \Rightarrow \beta) \equiv(\neg \beta \Rightarrow-\alpha)\)
(D) \((\neg (\alpha \vee \beta)) \equiv(\neg \alpha \Rightarrow-\beta)\)
Choose the correct answer from the options given below :
Answer: C. (A) and (C) only — Conclusion: The distributive equivalence (α ∧ (β ∨ γ)) ≡ ((α ∧ β) ∨ (α ∧ γ)) and the contrapositive equivalence (α ⇒ β) ≡ (¬β ⇒ ¬α) are correct; the other two…
- A.
(A) and (D) only
- B.
(B) and (C) only,
- C.
(A) and (C) only
- D.
(B) and (D) only
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Correct answer: C
Conclusion: The distributive equivalence (α ∧ (β ∨ γ)) ≡ ((α ∧ β) ∨ (α ∧ γ)) and the contrapositive equivalence (α ⇒ β) ≡ (¬β ⇒ ¬α) are correct; the other two given formulas are not equivalent.
Distributive law: (α ∧ (β ∨ γ)) ≡ ((α ∧ β) ∨ (α ∧ γ)). This is a standard logical distributive law: requiring α and at least one of β or γ is equivalent to either (α and β) or (α and γ).
(α ∨ β) ≡ (¬α ∨ β) is false. The formula ¬α ∨ β is equivalent to (α ⇒ β), not to α ∨ β. Counterexample: take α = true, β = false. Then α ∨ β is true but ¬α ∨ β is false, so they are not equivalent.
(α ⇒ β) ≡ (¬β ⇒ ¬α). This is the contrapositive equivalence: an implication is always equivalent to its contrapositive.
¬(α ∨ β) ≡ (¬α ⇒ ¬β) is false.By De Morgan, ¬(α ∨ β) is equivalent to ¬α ∧ ¬β, not to an implication. Counterexample: take α = true and β = false. Then ¬(α ∨ β) is false, while (¬α ⇒ ¬β) is true (since the antecedent ¬α is false), so they differ.
Therefore the two correct equivalences are the distributive law (the first formula above) and the contrapositive equivalence (the implication and its contrapositive).
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