Let E be the set of rational numbers in the interval [0, 1]. Let XE be the characteristic function of the set E 1; 0; x is irrational x is rational XE(x) = { {0,1/n, 2/n, · - · , n/n} of [0,1]. Show that XE is not R. I. Hint: take the partition Pn What is Up, (xE) and Lp, (XE) (recall that every nonempty interval must contain rational and irrational numbers)? Now take the limit of UP, (XE) and Lp,(XE) as n →0o. Conclude that yp is not R. I. ...
Let E be the set of rational numbers in the interval [0, 1]. Let XE be the characteristic function of the set E 1; 0; x is irrational x is rational XE(x) = { {0,1/n, 2/n, · - · , n/n} of [0,1]. Show that XE is not R. I. Hint: take the partition Pn What is Up, (xE) and Lp, (XE) (recall that every nonempty interval must contain rational and irrational numbers)? Now take the limit of UP, (XE) and Lp,(XE) as n →0o. Conclude that yp is not R. I. ...
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 E be the set of rational numbers in the interval [0, 1]. Let XE be the characteristic
function of the set E
x is rational
xe(#) = {
1;
0; x is irrational
{0, 1/n, 2/n, ..., n/n} of [0,1].
Show that XE is not R. I. Hint: take the partition Pn
What is UP, (xXE) and LP,(Xe) (recall that every nonempty interval must contain rational
and irrational numbers)? Now take the limit of UP, (XE) and LP, (XE) as n → 0. Conclude
that XE is not R. I.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed14a3ea-da26-4be7-a143-8b845df95e91%2Fff1d5e47-e916-42de-b7b7-39f3e61297d1%2Fzbx6lpd_processed.png&w=3840&q=75)
Transcribed Image Text:Let E be the set of rational numbers in the interval [0, 1]. Let XE be the characteristic
function of the set E
x is rational
xe(#) = {
1;
0; x is irrational
{0, 1/n, 2/n, ..., n/n} of [0,1].
Show that XE is not R. I. Hint: take the partition Pn
What is UP, (xXE) and LP,(Xe) (recall that every nonempty interval must contain rational
and irrational numbers)? Now take the limit of UP, (XE) and LP, (XE) as n → 0. Conclude
that XE is not R. I.
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