The statement \((¬p)⇒(¬q)\) is logically equivalent to which of the statements…
2017
The statement \((¬p)⇒(¬q)\) is logically equivalent to which of the statements below?
I. \(p \Rightarrow q\)
II. \(q \Rightarrow p\)
III. \(\left ( ¬q \right ) \vee p\)
IV. \(\left ( ¬p \right ) \vee q\)
- A.
I only
- B.
I and IV only
- C.
II only
- D.
II and III only
Attempted by 221 students.
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Correct answer: D
Key idea: rewrite each implication as a disjunction using a ⇒ b ≡ ¬a ∨ b.
Start with the given statement: (¬p) ⇒ (¬q).
Rewrite it as a disjunction: (¬p) ⇒ (¬q) ≡ ¬(¬p) ∨ (¬q) = p ∨ ¬q.
Compare with the listed statements:
q ⇒ p is ¬q ∨ p, which is the same as p ∨ ¬q.
The disjunction (¬q) ∨ p is exactly p ∨ ¬q as well.
Conclusion: The original statement is equivalent to q ⇒ p and to (¬q) ∨ p. The other listed form p ⇒ q (which is ¬p ∨ q) is not equivalent.
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