GATE CS 2017 Set 1 — Question 1
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Engineering Mathematics → Mathematical Logic → Logical Equivalence
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Question
The statement is logically equivalent to which of the statements below?I.
II.
III.
IV.
II.
III.
IV.
Correct answer
(D) II and III only
Solution
The given statement is .
Statement IV () is equivalent to , not the original statement.Thus, statements II and III are logically equivalent to the given statement.
1.Contrapositive: The contrapositive of an implication is , which is logically equivalent to the original statement.
- Contrapositive of is , which simplifies to .
- This matches statement II.
- Applying this to , we get , which simplifies to .
- Since disjunction is commutative, is equivalent to .
- This matches statement III.
Statement IV () is equivalent to , not the original statement.Thus, statements II and III are logically equivalent to the given statement.
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