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Let SCR. Give counterexamples to show that the following statements are not true. (a) If S is a bounded collection of rational numbers, then sup SE Q. (b) If S is closed, then sup SES and inf SES. (c) If S is open, then S is not closed. (d) The union of infinitely many closed sets is closed. (e) The intersection of infinitely many open sets is open.

Let SCR. Give counterexamples to show that the following statements are not true. (a) If S is a bounded collection of rational numbers, then sup SE Q. (b) If S is closed, then sup SES and inf SES. (c) If S is open, then S is not closed. (d) The union of infinitely many closed sets is closed. (e) The intersection of infinitely many open sets is open.

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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Let S be a subset of R. Give counterexamples to show that the following statements are not true.

 

Let SCR. Give counterexamples to show that the following statements are not true. (a) If S is a bounded collection of rational numbers, then sup S E Q. (b) If S is closed, then sup SES and inf S S. (c) If S is open, then S is not closed. (d) The union of infinitely many closed sets is closed. (e) The intersection of infinitely many open sets is open.
Transcribed Image Text:Let SCR. Give counterexamples to show that the following statements are not true. (a) If S is a bounded collection of rational numbers, then sup S E Q. (b) If S is closed, then sup SES and inf S S. (c) If S is open, then S is not closed. (d) The union of infinitely many closed sets is closed. (e) The intersection of infinitely many open sets is open.
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