Which of the following are logically equivalent ? A. \(\neg p \rightarrow (q…
2021
Which of the following are logically equivalent ?
A. \(\neg p \rightarrow (q \rightarrow r) \ and \ q \rightarrow (p \lor r) \)
B. \(( p \rightarrow q ) \rightarrow r \ and \ p \rightarrow ( q \rightarrow r ) \)
C. \(( p \rightarrow q ) \rightarrow ( r \rightarrow s ) \ and \ ( p \rightarrow r ) \rightarrow ( q \rightarrow s ) \)
Choose the correct answer from the options given :
- A.
A only
- B.
A and B only
- C.
B and C only
- D.
A and C only
Attempted by 90 students.
Show answer & explanation
Correct answer: A
We test each given pair for logical equivalence by rewriting implications as disjunctions (a → b ≡ ¬a ∨ b) and, where helpful, giving a counterexample.
Pair: ¬p → (q → r) and q → (p ∨ r).
Rewrite:
¬p → (q → r) ≡ p ∨ ¬q ∨ r
q → (p ∨ r) ≡ ¬q ∨ p ∨ r
Conclusion: These simplify to the same disjunction (p ∨ ¬q ∨ r), so this pair is logically equivalent.
Pair: (p → q) → r and p → (q → r).
Rewrite:
(p → q) → r ≡ (p ∧ ¬q) ∨ r
p → (q → r) ≡ ¬p ∨ ¬q ∨ r
Counterexample:
Take p = false, q = true, r = false. Then (p → q) → r evaluates to false, while p → (q → r) evaluates to true. Hence these formulas are not equivalent.
Pair: (p → q) → (r → s) and (p → r) → (q → s).
Rewrite:
(p → q) → (r → s) ≡ (p ∧ ¬q) ∨ ¬r ∨ s
(p → r) → (q → s) ≡ (p ∧ ¬r) ∨ ¬q ∨ s
Counterexample:
Take p = false, q = true, r = false, s = false. Then (p → q) → (r → s) is true while (p → r) → (q → s) is false. Therefore these formulas are not equivalent.
Final conclusion: Only the pair consisting of ¬p → (q → r) and q → (p ∨ r) are logically equivalent; the other two pairs are not.
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