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On the set of all integers Z, a relation R is defined by R = {(a, b): a - b is divisible by 2}. Which of the following best describes R?

1. R is an equivalence relation (reflexive, symmetric and transitive)
2. R is reflexive and symmetric but not transitive
3. R is symmetric and transitive but not reflexive
4. R is reflexive only

Correct answer: R is an equivalence relation (reflexive, symmetric and transitive)

Solution

R relates two integers when their difference is even, i.e. when both are even or both are odd (same parity). A relation that is reflexive, symmetric and transitive is an equivalence relation, so each property must hold for all integers in Z.

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