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Consider the relation R on the set {1, 2, 3} defined by R = {(1,1), (2,2), (3,3), (1,2), (2,3)}. Which of the following correctly describes R?

1. Reflexive, but neither symmetric nor transitive
2. Reflexive and symmetric, but not transitive
3. Reflexive and transitive, but not symmetric
4. An equivalence relation

Correct answer: Reflexive, but neither symmetric nor transitive

Solution

Reflexive: (1,1), (2,2), (3,3) are all present, so R is reflexive. Symmetric: (1,2) is in R but (2,1) is not, so R is not symmetric. Transitive: (1,2) and (2,3) are in R, but (1,3) is not, so R is not transitive. Hence R is reflexive but neither symmetric nor transitive.

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