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in Mathematics by (52.5k points)

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.

1 Answer

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Best answer

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) ∈ R

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