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A. Verify that the Manhattan metric (from page 71) is really a metric on M = R². (The interesting part is checking the triangle inequality.)

A. Verify that the Manhattan metric (from page 71) is really a metric on M = R². (The interesting part is checking the triangle inequality.)

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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A. Verify that the Manhattan metric (from page 71) is really a metric on M = R². (The interesting part is checking the triangle inequality.)
Transcribed Image Text:A. Verify that the Manhattan metric (from page 71) is really a metric on M = R². (The interesting part is checking the triangle inequality.)
Example B: Let M=IR², and let x=(x₁, x₂), y= (y₁, ye). Then d₂ = √√√(x₁-x₁)² + (x₂ - y₂)²² is the Euclidean metue is the Manhattan metna d₁= 1x₁-y₁1 + 1x₂-y₂l and do= max {1x₁-y₁l, 1x₂-yel} is the max metre. All three () are actually metrics, do is a metric, as: • dlx, x)=0, while if x=y then they differ in at least one cardinate, so dozu. ● 0 dlx, y)=dly, x) is immediate from the definition max { 1x₁-2₁1, 1x₂-2₂1} <mx{1x₁-y, +v.-2.1, 1x-very-al} (by inquality on IR) • is ≤d₂(x, y) + d₂(y, z) by inspection of cases, 1) da is 2) d₂ is and this metric, Self-checki Verify the A-inequality, a a metriz, C Sepuratin + symmetry are immedink, rig the since it is identical to the norm metni usual identification of & with IR².
Transcribed Image Text:Example B: Let M=IR², and let x=(x₁, x₂), y= (y₁, ye). Then d₂ = √√√(x₁-x₁)² + (x₂ - y₂)²² is the Euclidean metue is the Manhattan metna d₁= 1x₁-y₁1 + 1x₂-y₂l and do= max {1x₁-y₁l, 1x₂-yel} is the max metre. All three () are actually metrics, do is a metric, as: • dlx, x)=0, while if x=y then they differ in at least one cardinate, so dozu. ● 0 dlx, y)=dly, x) is immediate from the definition max { 1x₁-2₁1, 1x₂-2₂1} <mx{1x₁-y, +v.-2.1, 1x-very-al} (by inquality on IR) • is ≤d₂(x, y) + d₂(y, z) by inspection of cases, 1) da is 2) d₂ is and this metric, Self-checki Verify the A-inequality, a a metriz, C Sepuratin + symmetry are immedink, rig the since it is identical to the norm metni usual identification of & with IR².
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