2. Letde(x, y) = |x — y|, d₁(x, y) =for x, y E R. Show that(a) d₁ is a metric on R;(b) de and d₁ are not equivalent.|x - y|1 + x - y

Answered: Image /qna-images/answer/9ccb1871-aab2-4ca9-9916-53223a5b03c8.jpg

Answered: Let de(x, y) := |x — y|, d₁(x,y) := for…

Let de(x, y) := |x — y|, d₁(x,y) := for x,y E R. Show that (a) d₁ is a metric on R; (b) dẼ and d₁ are not equivalent. |x - y| 1 + x − y|

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.6: The Three-dimensional Coordinate System

Problem 41E: Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?

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Question

2. Let
de(x, y) = |x — y|, d₁(x, y) =
for x, y E R. Show that
(a) d₁ is a metric on R;
(b) de and d₁ are not equivalent.
|x - y|
1 + x - y

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Step 1: Define a theorem corresponding to equivalent metric

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Step 2: Show the dE is a metric on R

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Step 3: Show that d1 is a metric on R

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Step 4: Use inequality

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Step 5: Prove that d1 is a metric on R

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Step 6: Use theorem to check dE and d1 are equivalent or not

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Step 7: Use theorem

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