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Which of the following correctly represents the logical equivalence of [(p ∨ q) ∧ (¬q)] ∨ (¬p)?

1. The dual of [(p ∨ q) ∧ (¬q)] ∨ (¬p) is [(p ∧ q) ∨ (¬q)] ∧ (¬p).
2. The logical equivalence of [(p ∨ q) ∧ (¬q)] ∨ (¬p) is [(p ∧ ¬q) ∨ (q ∧ ¬q)] ∨ (¬p) or [(p ∧ ¬q) ∨ (¬p)].
3. The negation of [(p ∨ q) ∧ (¬q)] ∨ (¬p) is ¬[(p ∨ q) ∧ (¬q)] ∧ (¬p) or [(¬p ∧ ¬q) ∨ q] ∧ (¬p).
4. The contrapositive of [(p ∨ q) ∧ (¬q)] ∨ (¬p) is ¬[(p ∨ q) ∧ (¬q)] or ¬[(¬p ∧ ¬q) ∨ q].

Correct answer: The logical equivalence of [(p ∨ q) ∧ (¬q)] ∨ (¬p) is [(p ∧ ¬q) ∨ (q ∧ ¬q)] ∨ (¬p) or [(p ∧ ¬q) ∨ (¬p)].

Solution

The logical equivalence simplifies to [(p ∧ ¬q) ∨ (¬p)] because the distributive property and negation rules are applied to the original expression. This matches the given equivalence.

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