Let C₁ denote the sets defined in exercise 2. Let us also define, for each positive integer n, the closed interval En := [0,2]. (a) Show that for each 7, there exists some real number such that En = {n+y: for some y € Cn}. (b) Is the following statement true? There exists a real number r such that for each n, we have En = {x+y: for some y € Cn}.
Let C₁ denote the sets defined in exercise 2. Let us also define, for each positive integer n, the closed interval En := [0,2]. (a) Show that for each 7, there exists some real number such that En = {n+y: for some y € Cn}. (b) Is the following statement true? There exists a real number r such that for each n, we have En = {x+y: for some y € Cn}.
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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I've finished number 2, but I cannot figure out what number 3 is asking. Please explain and answer it. Thank you!
![2. For each positive integer n, consider the closed interval
C₂ :=
-
Similarly, consider the open intervals
(a) If N is a positive integer, express the set
as an interval.
(b) Let us define
as an interval.
(d) Define the set
TL TL
Dn:=(-n,n).
How many different elements belong to C?
Sidenote: This is how we can define intersection for infinitely many sets.
(c) If N is a positive integer, express the set
D₁ UD₂U... U DN
and
G₂n C₂ C₂n...n CN
C:= {1 € R: for each positive integer n, z
n, I € Cn}.
(b) Is the following statement true?
D:= {1ER:1€ Dn, for some n € N}.
Describe in an alternative way the set R\ D.
Sidenote: This is how we can define the union of infinitely many sets.
3. Let C₁, denote the sets defined in exercise 2. Let us also define, for each positive integer n, the closed
interval En == [0,2].
(a) Show that for each n, there exists some real number, such that
En = {n+y: for some y € Cn}-
There exists a real number such that for each n, we have
En = {1+y: for some y € Cn}-
Definition 1. Given a function f: R→ R, and given a set ACR and a set BCR, we define the image
of A and the pre-image of B as
f(A) := {f(1) = R: for some I € A},
f¹(B) := {1 € R: there exists y B with f(x) = y).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4893c46-a94c-4c81-a9fc-277ce07300a6%2F847e65d6-959f-4bce-924a-fd1ee52db5a0%2Fwzm84b5_processed.png&w=3840&q=75)
Transcribed Image Text:2. For each positive integer n, consider the closed interval
C₂ :=
-
Similarly, consider the open intervals
(a) If N is a positive integer, express the set
as an interval.
(b) Let us define
as an interval.
(d) Define the set
TL TL
Dn:=(-n,n).
How many different elements belong to C?
Sidenote: This is how we can define intersection for infinitely many sets.
(c) If N is a positive integer, express the set
D₁ UD₂U... U DN
and
G₂n C₂ C₂n...n CN
C:= {1 € R: for each positive integer n, z
n, I € Cn}.
(b) Is the following statement true?
D:= {1ER:1€ Dn, for some n € N}.
Describe in an alternative way the set R\ D.
Sidenote: This is how we can define the union of infinitely many sets.
3. Let C₁, denote the sets defined in exercise 2. Let us also define, for each positive integer n, the closed
interval En == [0,2].
(a) Show that for each n, there exists some real number, such that
En = {n+y: for some y € Cn}-
There exists a real number such that for each n, we have
En = {1+y: for some y € Cn}-
Definition 1. Given a function f: R→ R, and given a set ACR and a set BCR, we define the image
of A and the pre-image of B as
f(A) := {f(1) = R: for some I € A},
f¹(B) := {1 € R: there exists y B with f(x) = y).
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