Question

함수 가 집합 시상에서 거리함수이면 다음과 같이 정의된 한스) D: NN-R ALD) - + 3.0) (0) 함수 D는 시상에서 거리함수임을 보이시오. 1) 임의의 , VEN에 대하여 DI ) < 1이 됨을 보이시오.

If function d is a distance function on set x, then the function D defined as

D : X×X→R s.t. D(x, y)= d(x, y)/(1+d(x, y))

(1) Show that function D is a distance function on x
(2)Show that any x, y becomes D (x, y)<1

Answer 1

function Day = d(x,y) Hd (ay) To show that D is distance i Dory)= d (ay) - Itd(19) dince d is distance function d (3,7) 7,0 tony EX -) D(x, y) = (x,y) zo td Gyo Now Suppose D(x,y)=0 dlar,y) Hd (27) - d (ay) zo. x=y - (. dis distance function, D(x,y) = 0 iff azy : in) Day) = d (3,) ltday) d (y,x serday)=dlya) = D(y,x function)

Udle Page 1 for any a, J, Z EX ico day) s d (0,2) + d (4,7) aneعدل عنه:)۔ - Thamnom.) Day dlay & d(x, 2) td (Y,Z) Id(2, 2) + d(y, 2) - d (a, z) dly, z) It d (2, 2)+dly, z) Adlaza de I dla; 2) + d (), 2) = 1 + (42) _ Itd (972) - ED (3, 2) to (9, 2) Day) < D (or, 2) + Dly, z) =) Triangle inegudlity follows = D is a distance function 2 Now 1 Day - dlay) itday)

Here day) 7,0 - dla,y) Kitday व(व,५) ।। ) =) (,9) yex