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Show that the Relation R in the set Z of integers given by R= {(x y) : 2 divides (x-y)} is an Equivalence Relation. |
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Answer» → R – {(xy): 2 divides (x - y)} → R is Reflexive as 2 divides (a - a) ∀ a ∈ z. If (a b) ∈ R then 2 divides a - b. 2 divides b - a Hence (ba) ∈ R ⇒R is Symmetric. (ab) ∈ R and (be) ∈ R ⇒ a - b and b-c are divisible by 2 Now a - c = (a - b) + (b - c) = is even ∴ a - c is divisible by 2. ⇒ R is Transitive R is Reflexive, Transitive, Symmetric ∴ R is an Equivalence Relation in Z. |
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