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Question

R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | x − y | ≤ 1. Then, R is
(a) reflexive and transitive
(b) reflexive and symmetric
(c) symmetric and transitive
(d) an equivalence relation

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Solution

(b) reflexive and symmetric

Reflexivity: Let xR. Then,x-x=0 < 1x-x1x, xR for all xZSo, R is reflexive on Z.Symmetry: Let x, yR. Then,x-y 0-(y-x) 1y-x 1 Since x-y=y-xy, xR for all x, yZSo, R is symmetric on Z.Transitivity: Let x, yR and y, zR. Then,x-y 1 and y-z 1It is not always true that x-y 1.x, zRSo, R is not transitive on Z.

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