3. Let E be the set of rational numbers in the interval (0, 1). Let XE be the characteristic function of the set E 1; r is rational 0; x is irrational Show that XE is not R. I. Hint: take the partition P, = {0,1/n, 2/n, .. , n/n} of [0, 1). 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 → 0. Conclude Xe(x) that XE is not R. I.
3. Let E be the set of rational numbers in the interval (0, 1). Let XE be the characteristic function of the set E 1; r is rational 0; x is irrational Show that XE is not R. I. Hint: take the partition P, = {0,1/n, 2/n, .. , n/n} of [0, 1). 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 → 0. Conclude Xe(x) that XE 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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Question
#3. Please see attached. Thanks.
![3. Let E be the set of rational numbers in the interval [0, 1]. Let XE be the characteristic
function of the set E
XE(x) = {
Į 1;
x is rational
0; x is irrational
Show that XE is not R. I. Hint: take the partition Pn = {0,1/n, 2/n, -..,n/n} of [0, 1].
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 → o. Conclude
that XE is not R. I.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F605dc75a-6667-48dc-a2a9-e65d2e5eff36%2Fsu33hlh_processed.png&w=3840&q=75)
Transcribed Image Text:3. Let E be the set of rational numbers in the interval [0, 1]. Let XE be the characteristic
function of the set E
XE(x) = {
Į 1;
x is rational
0; x is irrational
Show that XE is not R. I. Hint: take the partition Pn = {0,1/n, 2/n, -..,n/n} of [0, 1].
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 → o. Conclude
that XE is not R. I.
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