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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 |
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Answer» ∀ 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 |
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