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Let p, q, and r be arbitrary logical propositions. Which of the following equivalences is correct?

1. ~[p ∧ (~q)] ≡ (~p) ∧ q
2. ~[(p ∨ q) ∧ (~r)] ≡ (~p) ∨ (~q) ∨ (~r)
3. ~[p ∨ (~q)] ≡ (~p) ∧ q
4. ~[p ∨ (~q)] ≡ (~p) ∨ ~q

Correct answer: ~[p ∨ (~q)] ≡ (~p) ∧ q

Solution

The correct option is valid because it applies De Morgan's laws, which state that the negation of a disjunction is equivalent to the conjunction of the negations. Thus, negating the expression 'p or not q' results in 'not p and q', confirming the equivalence.

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