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
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 32EQ
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