Metric spaces 3. For a metric space (X, d), the diameter of a subset AC X is defined by8(A):= sup d(x, y).x,yƐAHere we allow 8(A)= ∞. Show that if A, B C X satisfy ANB + Ø, then 8(AU B) < 8(A) + 8(B). Givean example to show this is not necessarily true when AN B = Ø.

Answered: Image /qna-images/answer/be6acd0b-1cfa-4e9d-ad6e-ceb92bbf7a71.jpg

3. For a metric space (X, d), the diameter of a subset A C X is defined by 6(A) := sup d(x, y). x,yƐA Here we allow 8(A) = o. Show that if A, B C X satisfy AN B + 0, then 8(AUB) < 8(A) + 8(B). Give an example to show this is not necessarily true when ANB = Ø.

3. For a metric space (X, d), the diameter of a subset A C X is defined by 6(A) := sup d(x, y). x,yƐA Here we allow 8(A) = o. Show that if A, B C X satisfy AN B + 0, then 8(AUB) < 8(A) + 8(B). Give an example to show this is not necessarily true when ANB = Ø.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

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

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

Chapter8: Areas Of Polygons And Circles

Section8.CR: Review Exercises

Problem 38CR: Prove that if semicircles are constructed on each of the sides of a right triangle, then the area of...

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Question

Metric spaces

3. For a metric space (X, d), the diameter of a subset AC X is defined by
8(A)
:= sup d(x, y).
x,yƐA
Here we allow 8(A)
= ∞. Show that if A, B C X satisfy ANB + Ø, then 8(AU B) < 8(A) + 8(B). Give
an example to show this is not necessarily true when AN B = Ø.

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