Consider the sentences [math: ¬((p→¬q)∧r)] and [math: (p∧q)∨¬r]. There are 8 possible interpretations for these sentences according to the truth values of [math: p], [math: q] and [math: r]. Under how many of these interpretations do the two sentences have the same truth value?

Question

BlackTom AI · Accepted Answer

p → ¬q is equivalent to ¬p ∨ ¬q, and De Morgan’s laws let us move negations across conjunctions/disjunctions.

1. Relevant knowledge points and concepts
- Material implication: (A → B) ≡ (¬A ∨ B).
- De Morgan’s laws: ¬(X ∧ Y) ≡ (¬X ∨ ¬Y) and ¬(X ∨ Y) ≡ (¬X ∧ ¬Y).
- Double negation and propositional equivalence: replacing formulas by equivalent formulas preserves truth values in every interpretation.

2. Complete reasoning process (step by step)
Start with the first sentence: ¬((p → ¬q) ∧ r).
- Replace the implication using material implication:
  p → ¬q ≡ ¬p ∨ ¬q.
  So ¬((p → ¬q) ∧ r) ≡ ¬((¬p ∨ ¬q) ∧ r).
- Apply De Morgan’s law to the negated conjunction:
  ¬((¬p ∨ ¬q) ∧ r) ≡ ¬(¬p ∨ ¬q) ∨ ¬r.
- Apply De Morgan’s law again to ¬(¬p ∨ ¬q):
  ¬(¬p ∨ ¬q) ≡ p ∧ q.
- Substitute back:
  ¬((p → ¬q) ∧ r) ≡ (p ∧ q) ∨ ¬r.

3. Intermediate steps shown
- p → ¬q ≡ ¬p ∨ ¬q
- ¬((¬p ∨ ¬q) ∧ r) ≡ ¬(¬p ∨ ¬q) ∨ ¬r
- ¬(¬p ∨ ¬q) ≡ p ∧ q
- Therefore ¬((p → ¬q) ∧ r) ≡ (p ∧ q) ∨ ¬r

4. Why the correct answer is correct
The two formulas are logically equivalent (syntactically reduced to the same formula). Logical equivalence means they have identical truth values under every valuation of p, q, r. Since there are 2^3 = 8 possible truth assignments, they agree on all 8.

5. Why other counts would be incorrect
Any answer smaller than 8 would assert there exists at least one valuation where the formulas differ; the algebraic equivalence above rules that out. Thus counts like 0–7 are incorrect.

6. Conclusion and confirmation
Because the two sentences are logically equivalent, they have the same truth value in all 8 interpretations. Final answer: 8.

Consider the sentences [math: ¬((p→¬q)∧r)] and [math: (p∧q)∨¬r]. There are 8 possible interpretations for these sentences according to the truth values of [math: p], [math: q] and [math: r]. Under how many of these interpretations do the two sentences have the same truth value?简答题

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