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4.4. Let S be a nonempty set. Suppose that to each ordered pair (x, y) E S x S a nonnegative real number d(x, y) is assigned that satisfies the following conditions: (i). d(x, y) > 0 and d(x, y) = 0 if and only if r y, (ii). d(x, y) = d(y, x) for all a, y E S, %3D (iii). d(x, z) < d(x, y) + d(y, z) for all a, y, z E S. Then, d(x, y) is called a metric on S. The set S with metric d is called a metric space and is denoted as (S, d). Show that in C the following are metrics: (a). d(z, w) = |2 – wl, |z – w| 1+|2-삐' S O if z = w | 1 if z # w. (b). d(z, w): %3D (c). d(z, w) = %3D

4.4. Let S be a nonempty set. Suppose that to each ordered pair (x, y) E S x S a nonnegative real number d(x, y) is assigned that satisfies the following conditions: (i). d(x, y) > 0 and d(x, y) = 0 if and only if r y, (ii). d(x, y) = d(y, x) for all a, y E S, %3D (iii). d(x, z) < d(x, y) + d(y, z) for all a, y, z E S. Then, d(x, y) is called a metric on S. The set S with metric d is called a metric space and is denoted as (S, d). Show that in C the following are metrics: (a). d(z, w) = |2 – wl, |z – w| 1+|2-삐' S O if z = w | 1 if z # w. (b). d(z, w): %3D (c). d(z, w) = %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4.4. Let S be a nonempty set. Suppose that to each ordered pair (x, y) E S x S a nonnegative real number d(x, y) is assigned that satisfies the following conditions: (i). d(x, y) >0 and d(x, y) = 0 if and only if r = y, (ii). d(x, y) = d(y, x) for all r, y E S, (iii). d(x, 2) < d(x, y) + d(y, z) for all x, y, z E S. Then, d(x, y) is called a metric on S. The set S with metric d is called a metric space and is denoted as (S, d). Show that in C the following are metrics: (a). d(z, w) = |2 – w|, |z – w| 1+|z – w|' (b). d(z, w) = ( O if z = w | 1 if z + w. (c). d(z, w) :
Transcribed Image Text:4.4. Let S be a nonempty set. Suppose that to each ordered pair (x, y) E S x S a nonnegative real number d(x, y) is assigned that satisfies the following conditions: (i). d(x, y) >0 and d(x, y) = 0 if and only if r = y, (ii). d(x, y) = d(y, x) for all r, y E S, (iii). d(x, 2) < d(x, y) + d(y, z) for all x, y, z E S. Then, d(x, y) is called a metric on S. The set S with metric d is called a metric space and is denoted as (S, d). Show that in C the following are metrics: (a). d(z, w) = |2 – w|, |z – w| 1+|z – w|' (b). d(z, w) = ( O if z = w | 1 if z + w. (c). d(z, w) :
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