Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use the fact that x- z| = |x+y-y-zs |x-y\ + ly-z| to show part ii).

Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use the fact that x- z| = |x+y-y-zs |x-y\ + ly-z| to show part ii).

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
Definition 4.2.1. Suppose S is a non-empty set. A metric d is a function d:S x S [0, 0) such that the following hold: i). d(a, y) = 0 if and only if æ = y ii). d(x, y) = d(y, æ) for all æ and y in S. iii). d(x, z) < d(x, y) + d(y, z) for all a, y, z in S. Show that d given by d(x, y) = x-y is a metric on IR by using definition 4.2.1. You may want to use the fact that x-z|= |x+y-y-zs |x-y + ly-z| to show part ii).
Transcribed Image Text:Definition 4.2.1. Suppose S is a non-empty set. A metric d is a function d:S x S [0, 0) such that the following hold: i). d(a, y) = 0 if and only if æ = y ii). d(x, y) = d(y, æ) for all æ and y in S. iii). d(x, z) < d(x, y) + d(y, z) for all a, y, z in S. Show that d given by d(x, y) = x-y is a metric on IR by using definition 4.2.1. You may want to use the fact that x-z|= |x+y-y-zs |x-y + ly-z| to show part ii).
Expert Solution
Step 1

we knew  that given metric d(x,y) = |x-y| is always ≥0.

 now d(x,y) = 0 ⇔ |x-y| = 0

                         ⇔x-y = 0

                          ⇔x = y

hence (i) condition of 4.2.1 holds that d(x,y) = 0 if and only id x = y.

let x,y ∈ R be two arbitrary points.

 then d(x,y) = |x-y| = |y-x| = d(y,x)

hence condition (ii) is satishfied  that  d(x,y) = d(y,x) ,∀ x,y ∈ R

 

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Relations
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
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