Let π be the relation on the set of positive integers such that ππ π if andβ¦
2015
Let π be the relation on the set of positive integers such that ππ πΒ if and only if π and π are distinct and have a common divisor other than 1. Which one of the following statements about π is true?
- A.
π is symmetric and reflexive but not transitive
- B.
π is reflexive but not symmetric and not transitive
- C.
π is transitive but not reflexive and not symmetric
- D.
π is symmetric but not reflexive and not transitive
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Correct answer: D
Key properties:
Symmetric: If a and b are distinct and have a common divisor greater than 1, then b and a also are distinct and have the same common divisor. Thus aRb implies bRa.
Not reflexive: Reflexivity would require aRa for every positive integer a, but the definition requires a and b to be distinct, so aRa never holds.
Not transitive: Provide a counterexample. Take a = 2, b = 6, c = 3. Then 2 and 6 share the divisor 2 (so 2R6), and 6 and 3 share the divisor 3 (so 6R3), but 2 and 3 have no common divisor greater than 1 (so 2R3 is false). Hence transitivity fails.
Conclusion: R is symmetric but not reflexive and not transitive.
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