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Check whether the relation R in the set Z of integers defined as R = {(a,b): a + b is "divisible by 2"} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0]. |
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Answer» Reflexive : Since, a + a = 2a which is even ∴ (a,a) ∈ R ∀a ∈ Z Hence R is reflexive Symmetric: If (a,b) ∈ R, then a + b = 2λ ⇒ b + a = 2λ ⇒ (b,a) ∈ R, Hence R is symmetric Transitive: If (a,b) ∈ R and (b,c,) ∈ R Then a + b = 2λ ---(1) and b + c = 2μ --- (2) Adding (1) and (2) we get a + 2b + c = 2(λ + μ) ⇒ a + c = 2(λ + μ − b) ⇒ a + c = 2k, where λ + μ − b = k ⇒ (a,c) ∈ R Hence R is transitive [0] = {...-4, -2, 0, 2, 4...} |
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