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Problem 4. Let (X, d) be a metric space, and A, B C X. (a) Show that AUB AUB. (b) Similarly as in (a), show that for any family of sets (A;)ieI, we have Give an example in (R, |.|) to show that the other inclusion doesn't hold in general. (c) Show that ANBCANB. (d) Find examples of nonempty A, B in (R, |-|) such that A NB = 0, yet A inclusion in (c) can be strict!!) B. (In particular, the

Problem 4. Let (X, d) be a metric space, and A, B C X. (a) Show that AUB AUB. (b) Similarly as in (a), show that for any family of sets (A;)ieI, we have Give an example in (R, |.|) to show that the other inclusion doesn't hold in general. (c) Show that ANBCANB. (d) Find examples of nonempty A, B in (R, |-|) such that A NB = 0, yet A inclusion in (c) can be strict!!) B. (In particular, the

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
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Please do parts b and d

Problem 4. Let (X,d) be a metric space, and A, B C X. (a) Show that AUB=AU B. (b) Similarly as in (a), show that for any family of sets (A;)ieI, we have Give an example in (R, |.|) to show that the other inclusion doesn't hold in general. (c) Show that ANBCAN B. (d) Find examples of nonempty A, B in (R, |.|) such that An B = 0, yet A = B. (In particular, the inclusion in (c) can be strict!)
Transcribed Image Text:Problem 4. Let (X,d) be a metric space, and A, B C X. (a) Show that AUB=AU B. (b) Similarly as in (a), show that for any family of sets (A;)ieI, we have Give an example in (R, |.|) to show that the other inclusion doesn't hold in general. (c) Show that ANBCAN B. (d) Find examples of nonempty A, B in (R, |.|) such that An B = 0, yet A = B. (In particular, the inclusion in (c) can be strict!)
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