. 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?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 18E

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Question

The question and the options are attached in the following images. Please provide a proof for the correct answer.

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

C.
0, if |r – y| < 1
1, if otherwise
d(x, y) =
D. d(r, y) = |x – y|

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