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Let N be the set of natural numbers greater than 100. Define a relation R on N by: R = {(x, y) in N x N: x and y have at least two common divisors}. Which of the following properties does R possess?

1. Reflexive
2. Symmetric
3. Transitive
4. Equivalence relation

Correct answer: Reflexive

Solution

R is reflexive (every x shares 1 and x with itself) and symmetric (common divisors of x,y are same as y,x). But R is not transitive: for example x=6k and y=6m share divisors 1,2,3,6; y=6m and z=10m share divisors 1,2; x=6k and z=10m may share only 1 and 2. So not equivalence.

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