Question

consider the relation R on rational numbers defined by R={(a, b):a, b€Q, a-b€integers. show that, (a, b)€R implies that (b, a)for all a, b€Q​

Answer 1

Answer:

(i) Since a−a=0 and 0 is an even integer

(a,a)∈R

∴ R is reflexive.

(ii) If (a−b) is even, then (b−a) is also even. then, if (a−b)∈R,(b,a)∈R

∴ The relation is symmetric.

(iii) If (a,b)∈R,(b,c)∈R, then (a−b) is even, (b−c) is even, then $$(a-b

+b-c)=a-c$$ is even.

∴ If (a,b)∈R,(b,c)∈R implies (a,c)∈R

∴ R is transitive.

Since R is reflexive, symmetric and transitive, it is an equivalence relation.