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Ques-2-Metric-Space-Question-Let-d-be-a-metric-on-a-non-empty-set-X-Show-that-the-function-U-is-defined-by-U-x-y-d-x-y-1-d-x-y-where-x-and-y-are-arbitrary-element-X-is-also-a-metri




Question Number 191787 by Mastermind last updated on 30/Apr/23
Ques. 2 (Metric Space Question)        Let d be a metric on a non−empty  set X. Show that the function U is  defined by U(x,y)=((d(x,y))/(1+d(x,y))), where  x and y are arbitrary element X is also  a metric on X.
$$\mathrm{Ques}.\:\mathrm{2}\:\left(\mathrm{Metric}\:\mathrm{Space}\:\mathrm{Question}\right) \\ $$$$\:\:\:\:\:\:\mathrm{Let}\:\mathrm{d}\:\mathrm{be}\:\mathrm{a}\:\mathrm{metric}\:\mathrm{on}\:\mathrm{a}\:\mathrm{non}−\mathrm{empty} \\ $$$$\mathrm{set}\:\mathrm{X}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\:\mathrm{U}\:\mathrm{is} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\mathrm{U}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{d}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{1}+\mathrm{d}\left(\mathrm{x},\mathrm{y}\right)},\:\mathrm{where} \\ $$$$\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{are}\:\mathrm{arbitrary}\:\mathrm{element}\:\mathrm{X}\:\mathrm{is}\:\mathrm{also} \\ $$$$\mathrm{a}\:\mathrm{metric}\:\mathrm{on}\:\mathrm{X}. \\ $$

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