the following distance function is a metric on X. Let x = = (x₁, x₂), y = (91, 92) and z = (21, 22) (i).d(x, y) = √√√√(x₁ − y₁)² + (x2 − y2)² (Hint: Use the Minkowski's Inequality: √(an+bn)² ≤√√Σan² + √b² where an, bn are real numbers. x₁ - y₁| + x2 - y₂| (ii). d(x, y) = (iii). d(x, y) = max {|x1 - y₁|, |x2 - y2|}. 3. Extend each of the above three distance functions to define on X = R" and show that in each case the distance function is a metric.
the following distance function is a metric on X. Let x = = (x₁, x₂), y = (91, 92) and z = (21, 22) (i).d(x, y) = √√√√(x₁ − y₁)² + (x2 − y2)² (Hint: Use the Minkowski's Inequality: √(an+bn)² ≤√√Σan² + √b² where an, bn are real numbers. x₁ - y₁| + x2 - y₂| (ii). d(x, y) = (iii). d(x, y) = max {|x1 - y₁|, |x2 - y2|}. 3. Extend each of the above three distance functions to define on X = R" and show that in each case the distance function is a metric.
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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Please solve number 3
Transcribed Image Text:2. For each of the following, X represents Rx R. Prove that each of
the following distance function is a metric on X. Let x = (x₁, x₂), y =
(91, 92) and z (21, 22)
=
(i).d(x, y) = √(x₁ - y₁)² + (x2 − y2)²
(Hint: Use the Minkowski's Inequality:
Σ(an + bn)² ≤ √Σan² + √b² where an, bn are real numbers.
(ii). d(x, y) =
(iii). d(x, y) = max {x1y₁, x2 - y2|}.
-
x₁y₁| + x2 - y₂|
-
3. Extend each of the above three distance functions to define on
XR and show that in each case the distance function is a metric.
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