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(a) Prove that ANBCANB.Give an example to show that equality does not always hold.

(a) Prove that ANBCANB.Give an example to show that equality does not always hold.

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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How do you prove (a)?

Note that the bar represents the closure of a set

Def. 6 (Closure) We say that x belongs to the closure of S, and we write x E S, if x is in the interior or the boundary, that is, S = Sint U aS. This is the same as saying that S = {x E R" : Ve > 0, B(x; €, )NS+0}.
Transcribed Image Text:Def. 6 (Closure) We say that x belongs to the closure of S, and we write x E S, if x is in the interior or the boundary, that is, S = Sint U aS. This is the same as saying that S = {x E R" : Ve > 0, B(x; €, )NS+0}.
Let A and Bdenote subsets of R". (a) Prove that ANBCAN B. Give an example to show that equality does not always hold. (b) Prove that A° U B° C (AU B)°. Give an example to show that equality does not always hold. (c) Give an example of a function that is C2 but not C³.
Transcribed Image Text:Let A and Bdenote subsets of R". (a) Prove that ANBCAN B. Give an example to show that equality does not always hold. (b) Prove that A° U B° C (AU B)°. Give an example to show that equality does not always hold. (c) Give an example of a function that is C2 but not C³.
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