Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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
See similar textbooks
icon
Related questions
Question
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.
Expert Solution
steps

Step by step

Solved in 4 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS