Question

Let R: Z → Z be a relation defined by R = {(a, b) : a, b, ∈ Z, a – b ∈ z}. Show that 

(i) ∀ a ∈ Z, (a, a) ∈ R. 

(ii) (a, b) ∈ R ⇒ (b,a) ∈ R 

(iii) (a, b) ∈ R, (b, c) ∈ R ⇒ (a, c) ∈ R

Answer 1

∀ a ∈ Z, (a, a) ∈ R since a – a = 0 ∈ Z 

(a, b) ∈ R ⇒ a – b ∈ Z ⇒ ∴ b – a ∈ Z ⇒ (b, a) ∈ R 

(a, b) ∈ R, (b, c) ∈ R ⇒ a – b ∈ Z, b – C ∈ Z 

∴ a – b + b – c ∈ Z ⇒ (a, c) ∈ Z