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Question Number 119757 by Ar Brandon last updated on 26/Oct/20

For any integer n, let I_n  be the interval (n, n+1).  Define         R={(x, y)∈R∣both x, y ∈ I_n  for some n∈Z}  Then R is  (A) reflexive on R  (B) symmetric  (C) transitive  (D) an equivalence relation

$$\mathrm{For}\:\mathrm{any}\:\mathrm{integer}\:{n},\:\mathrm{let}\:{I}_{{n}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{interval}\:\left({n},\:{n}+\mathrm{1}\right). \\ $$$$\mathrm{Define} \\ $$$$\:\:\:\:\:\:\:\mathcal{R}=\left\{\left(\mathrm{x},\:\mathrm{y}\right)\in\mathbb{R}\mid\mathrm{both}\:\mathrm{x},\:\mathrm{y}\:\in\:{I}_{{n}} \:\mathrm{for}\:\mathrm{some}\:{n}\in\mathbb{Z}\right\} \\ $$$$\mathrm{Then}\:\mathcal{R}\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{reflexive}\:\mathrm{on}\:\mathbb{R} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{symmetric} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{transitive} \\ $$$$\left(\mathrm{D}\right)\:\mathrm{an}\:\mathrm{equivalence}\:\mathrm{relation} \\ $$

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