. Let R be the set of real numbers and let D be the set of functions d : R x R → R that satisfy the following properties for all r, y, z € R: • d(x, y) > 0 • d(x, y) = 0 if and only if r = y • d(r, y) = d(y, x) %3D d(x, z) < d(x, y) + d(y, z) Which of the following is not a function in D?
. Let R be the set of real numbers and let D be the set of functions d : R x R → R that satisfy the following properties for all r, y, z € R: • d(x, y) > 0 • d(x, y) = 0 if and only if r = y • d(r, y) = d(y, x) %3D d(x, z) < d(x, y) + d(y, z) Which of the following is not a function in D?
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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The question and the options are attached in the following images. Please provide a proof for the correct answer.
Transcribed Image Text:4. . Let R be the set of real numbers and let D be the set of functions d : R x R → R that
satisfy the following properties for all r, y, z E R:
• d(x, y) > 0
• d(r, y) = 0 if and only if r = y
• d(x, y) = d(y, x)
• d(x, z) < d(x, y) + d(y, z)
Which of the following is not a function in D?
A. d(r, y) = min{|x – y|, 1}
В.
d(r, y) = {
0, if r = y l
1, if r + y S
Transcribed Image Text:C.
0, if |r – y| < 1
1, if otherwise
d(x, y) =
D. d(r, y) = |x – y|
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