Question

let R be a relation on a defined by Q = { (a,b) /a, b belongs to Q, a-b belongs to Z }show that R is an equivalence relation.
​

Answer 1

Answer:

Let R be a relation from Q to Q defined by R = {(a, b)\ a,b ∈ Q and a – b ∈ Z}. Show that (i) (a, a) g R, for all a ∈ Q (ii) (a, b) ∈ R implies that (b, a) ∈R (iii) (a, b) ∈ R and (b, c) ∈R implies that (a, c) ∈ R.Read more on Sarthaks.com - https://www.sarthaks.com/583858/let-r-be-a-relation-from-q-to-q-defined-by-r-a-b-a-b-q-and-a-b-z-show-that-i-a-a-g-r-for-all-a-q

Given R = {{a, b): a,b∈Q and a-b ∈ Z) (i) ∀ a ∈ Q, a – a = 0 ∈ Z ⇒ (a, a) ∈ R (ii) Let (a, b) ∈ R⇒ a- b ∈ Z b – a ∈ Z ⇒ (b, a) ∈ R (iii) Let (a, b) and (b, c) g R ⇒ a – b ∈ Z and b – c ∈ Z a- c = (a - b) + (b – c) ∈ Z ∴ (a, c) ∈ RRead more on Sarthaks.com - https://www.sarthaks.com/583858/let-r-be-a-relation-from-q-to-q-defined-by-r-a-b-a-b-q-and-a-b-z-show-that-i-a-a-g-r-for-all-a-q