Question

State clearly the definition of a metric. Prove that R” equipped with the distance doo ((x1, ..., Un),(41, ... , Yn)) = max lli — yil, i=1,...,n is a metric.

Answer 1

That's easy, just follow the properties.

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Sola Definition of metric: Let X be a set. Then a function d: XRudol is said to be metric if it satis ties following: i) d(x,x) >0 HxEx ii) d (x,x) = 0 > x = 0 iii) d'(x,y) = diy, x) VXY EX iv) d (X, Z) 5d16, )+d19,2) WX,Y,ZEX, claim: do (x,y)= do ((X1,82,---, n), (41,42---, Yn)) doo (x,y) = max 121-47/ is a metric i) d (C1, 82, ---, xn) (9.42,---Yn)) = max 10i-4i120 This is since absolute value is always nonnegative ij) d[(x,x2, ---, Xb), (41,42, ---, Yn) = 0 s max 14;-yil = 0 > xi-48=0 Vi > x = 9 Vì Ø x=yi (x,162,--, xn) = (4,192,--, Yn) iii) d (x1,x2, ---, An),(41,42,--- Yn)) = Max 12-yil = wax 198-12 = d (14.92,--- Yn) (, ---, Hn)) iv) d{{x1,x2,---, An), (2,, Z2, ---, Zn)) Ewax 1 xi-zi / = max lxi-48 +49 – ZE / s max {1*9-4i/ +177-91} Max f18-991 + max 14i-3i|| = (12,-- an),(41,---,Yn))+d(14---Yn), (21.---> Zn)) From (i), (ii), (iii) prius doo is a metric on R?