Question

10. Definearelationon by setting x R y if x y is even.

(a) Give a counterexample to show that R is not reflexive.

(b) Give a counterexample to show that R is not transitive.

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References

10. Define a relation on Z by setting xRy if xy is even. (a) Give a counterexample to show that R is not reflexive (b) Give a counterexample to show that R is not transitive. 11. Explain why the web linking relation in Example 2.36 is not an equivalence relation. (You only need to give one reason why it fails to be an equivalence

Answer 1

A relation R is Reflexive if x R x for all x

If x x is even for all x

Let x = 3

3.3 = 9 which is not even

hence 3 is not related to 3 therefore R is not reflexive

A relation R is transitive if x R y and y R z then x R z

Let x= 3 , y =4 , z = 5

x y = 12 which is even therefore x R y ( 3 R 4 )

y z = 20 which is even therefore y R z ( 4 R 5 )

but x z = 3.5 = 15 which is not even hence x is not related to z ( 3 is not related to 5)

Therfore R is not transitive