Principles of Mathematics in Operations Research

Solutions 271

Problems of Chapter 10

10.1 Fix x,y eM.k arbitrary.

k k

d 2 (x,y) = \^2{xi - Vi)^2 ]1/2, di(x,y) = ^ |a?i -j/j|,

»=i i=i

doo{x,y) = max{|a;i - j/<|} = \XJ -%|.

i

di ~ doo-.

doo(x,y) = \XJ -yj\ < ^2\xi -yi\ = di(x,y) => A = 1.

»=i

doo(x,y) = |£j — 3/j | > \xi -yi\, Vi = 1,2, ...,/c

=>• kdocix^) = k\xj - yj\ > ^ |xi - i/j| => B = k.

[^(^y)]^2 = (XJ - yjf < ^2(xi - y{f => d^x.y) < d 2 {x,y) =>A = 1.

»=i

[doo(x,y)]2 = (a:, -1/,) >\2 \xt - yt\, Vi = 1,2, ...,fc

=>• fc[doo(a;, 2/)]^2 >M 2 (x,2/)]^2 ^S = Vfc.

di ~ c^: c?i ~ ^oo and d 2 ~ d<x> =>• <^i ~ ^2-

10.2

Consider the discrete metric dtp, a) = < '. ' on X.

Br(p) = {p} ,Br\p} = X, Br(P) = {p}? X.

10.3

(<=0

Let 0 ^ A C X. A is both open and closed. Let B = Ac, B is also both
open and closed. A U B = X. If A is closed then B is open, we have A (1B =
A n B = 0. If B is closed then .4 is open, we have Bni = 4nB = 0. Thus
X is disconnected.

(=*)
X is disconnected. 34 ^ 0,3B ^ 0 B X = AU B and (An B) n (Af) B) =
0 =» A n B = 0. Thus Ac = B ^0 =» A g X.
iUB=I=>v4UB = I, .4nB = 0=^vl = (B)c, i.e. ^ is open.