Mixing
Infinite union of countable sets proof.
$begingroup$
I understand how to prove that the union of 2 countable sets is countable. I then began to think we can use induction to say that the countable union of countable sets are also countable. However my textbook says otherwise. How come? Also it says that numbers arranged into a square like so
1 3 6 10 15 . . .
2 5 9 14 . . .
4 8 13 . . .
7 12 . . .
11 . . .
.
.
.
proves this theorem. I fail to make the connection.
Thanks alot.
real-analysis analysis
asked Jun 14 '16 at 8:22
benssiaobenssiao
113
$endgroup$
add a comment |
$begingroup$
I understand how to prove that the union of 2 countable sets is countable. I then began to think we can use induction to say that the countable union of countable sets are also countable. However my textbook says otherwise. How come? Also it says that numbers arranged into a square like so
1 3 6 10 15 . . .
2 5 9 14 . . .
4 8 13 . . .
7 12 . . .
11 . . .
.
.
.
proves this theorem. I fail to make the connection.
Thanks alot.
real-analysis analysis
asked Jun 14 '16 at 8:22
benssiaobenssiao
113
$endgroup$
add a comment |
$begingroup$
I understand how to prove that the union of 2 countable sets is countable. I then began to think we can use induction to say that the countable union of countable sets are also countable. However my textbook says otherwise. How come? Also it says that numbers arranged into a square like so
1 3 6 10 15 . . .
2 5 9 14 . . .
4 8 13 . . .
7 12 . . .
11 . . .
.
.
.
proves this theorem. I fail to make the connection.
Thanks alot.
real-analysis analysis
asked Jun 14 '16 at 8:22
benssiaobenssiao
113
$endgroup$
I understand how to prove that the union of 2 countable sets is countable. I then began to think we can use induction to say that the countable union of countable sets are also countable. However my textbook says otherwise. How come? Also it says that numbers arranged into a square like so
1 3 6 10 15 . . .
2 5 9 14 . . .
4 8 13 . . .
7 12 . . .
11 . . .
.
.
.
proves this theorem. I fail to make the connection.
Thanks alot.
real-analysis analysis
real-analysis analysis
asked Jun 14 '16 at 8:22
benssiaobenssiao
113
asked Jun 14 '16 at 8:22
benssiaobenssiao
113
edited Jun 14 '16 at 8:28
user99914
asked Jun 14 '16 at 8:22
benssiaobenssiao
113
asked Jun 14 '16 at 8:22
benssiaobenssiao
113
asked Jun 14 '16 at 8:22
benssiaobenssiao
113
113
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
An infinite union of countable sets may not be countable. For example, take the sets
$$A_x = {n+x| ninmathbb N}$$
so each set $A_x$ is countable, but $$bigcup_{xin[0,1]} A_x = mathbb R,$$
which is not countable.
However, your "square proof" would work fine to prove the statement
A countable union of countable sets is countable.
The proof would work because you can map the set $mathbb N$ to the countable union of countable sets, by mapping $1$ to the first element of the first set, then $2$ to the first element of the second set, $3$ to the first element of the first set, $4$ to the first element of the third set, and so on.
This statement would be very hard to prove by induction, however, because induction can only ever prove statements that are true for each element of $mathbb N$, not statements about the set as a whole.
answered Jun 14 '16 at 8:26
5xum5xum
91.8k394161
$endgroup$
$begingroup$
Awesome! Thanks alot. My textbook must have a type because it claims that an infinite union of countable sets is countable.
$endgroup$
– benssiao
Jun 14 '16 at 8:36
$begingroup$
@benssiao What book is that, and what page is the typo on, so we can all correct it in our copies?
$endgroup$
– bof
Jun 14 '16 at 8:52
$begingroup$
@benssiao Are you sure the textbook doesn't say "countably infinite"?
$endgroup$
– 5xum
Jun 14 '16 at 8:58
$begingroup$
Isnt countable and countably infinite the same? In the sense they are the same as saying a function is a bijection between N and A where A is any set. I have Stephen Abbott's second edition of Understanding Analysis. I am looking at theorem 1.5.8 ii on page 29.
$endgroup$
– benssiao
Jun 14 '16 at 9:06
$begingroup$
@benssiao Not entirely. Countable may also be finite. But even if you mean countable as "countably infinite", still, the terms infinite and countably infinite are certainly not the same.
$endgroup$
– 5xum
Jun 14 '16 at 9:09
|
show 2 more comments
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1 Answer
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1 Answer
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$begingroup$
An infinite union of countable sets may not be countable. For example, take the sets
$$A_x = {n+x| ninmathbb N}$$
so each set $A_x$ is countable, but $$bigcup_{xin[0,1]} A_x = mathbb R,$$
which is not countable.
However, your "square proof" would work fine to prove the statement
A countable union of countable sets is countable.
The proof would work because you can map the set $mathbb N$ to the countable union of countable sets, by mapping $1$ to the first element of the first set, then $2$ to the first element of the second set, $3$ to the first element of the first set, $4$ to the first element of the third set, and so on.
This statement would be very hard to prove by induction, however, because induction can only ever prove statements that are true for each element of $mathbb N$, not statements about the set as a whole.
answered Jun 14 '16 at 8:26
5xum5xum
91.8k394161
$endgroup$
$begingroup$
Awesome! Thanks alot. My textbook must have a type because it claims that an infinite union of countable sets is countable.
$endgroup$
– benssiao
Jun 14 '16 at 8:36
$begingroup$
@benssiao What book is that, and what page is the typo on, so we can all correct it in our copies?
$endgroup$
– bof
Jun 14 '16 at 8:52
$begingroup$
@benssiao Are you sure the textbook doesn't say "countably infinite"?
$endgroup$
– 5xum
Jun 14 '16 at 8:58
$begingroup$
Isnt countable and countably infinite the same? In the sense they are the same as saying a function is a bijection between N and A where A is any set. I have Stephen Abbott's second edition of Understanding Analysis. I am looking at theorem 1.5.8 ii on page 29.
$endgroup$
– benssiao
Jun 14 '16 at 9:06
$begingroup$
@benssiao Not entirely. Countable may also be finite. But even if you mean countable as "countably infinite", still, the terms infinite and countably infinite are certainly not the same.
$endgroup$
– 5xum
Jun 14 '16 at 9:09
|
show 2 more comments
$begingroup$
An infinite union of countable sets may not be countable. For example, take the sets
$$A_x = {n+x| ninmathbb N}$$
so each set $A_x$ is countable, but $$bigcup_{xin[0,1]} A_x = mathbb R,$$
which is not countable.
However, your "square proof" would work fine to prove the statement
A countable union of countable sets is countable.
The proof would work because you can map the set $mathbb N$ to the countable union of countable sets, by mapping $1$ to the first element of the first set, then $2$ to the first element of the second set, $3$ to the first element of the first set, $4$ to the first element of the third set, and so on.
This statement would be very hard to prove by induction, however, because induction can only ever prove statements that are true for each element of $mathbb N$, not statements about the set as a whole.
answered Jun 14 '16 at 8:26
5xum5xum
91.8k394161
$endgroup$
$begingroup$
Awesome! Thanks alot. My textbook must have a type because it claims that an infinite union of countable sets is countable.
$endgroup$
– benssiao
Jun 14 '16 at 8:36
$begingroup$
@benssiao What book is that, and what page is the typo on, so we can all correct it in our copies?
$endgroup$
– bof
Jun 14 '16 at 8:52
$begingroup$
@benssiao Are you sure the textbook doesn't say "countably infinite"?
$endgroup$
– 5xum
Jun 14 '16 at 8:58
$begingroup$
Isnt countable and countably infinite the same? In the sense they are the same as saying a function is a bijection between N and A where A is any set. I have Stephen Abbott's second edition of Understanding Analysis. I am looking at theorem 1.5.8 ii on page 29.
$endgroup$
– benssiao
Jun 14 '16 at 9:06
$begingroup$
@benssiao Not entirely. Countable may also be finite. But even if you mean countable as "countably infinite", still, the terms infinite and countably infinite are certainly not the same.
$endgroup$
– 5xum
Jun 14 '16 at 9:09
|
show 2 more comments
$begingroup$
An infinite union of countable sets may not be countable. For example, take the sets
$$A_x = {n+x| ninmathbb N}$$
so each set $A_x$ is countable, but $$bigcup_{xin[0,1]} A_x = mathbb R,$$
which is not countable.
However, your "square proof" would work fine to prove the statement
A countable union of countable sets is countable.
The proof would work because you can map the set $mathbb N$ to the countable union of countable sets, by mapping $1$ to the first element of the first set, then $2$ to the first element of the second set, $3$ to the first element of the first set, $4$ to the first element of the third set, and so on.
This statement would be very hard to prove by induction, however, because induction can only ever prove statements that are true for each element of $mathbb N$, not statements about the set as a whole.
answered Jun 14 '16 at 8:26
5xum5xum
91.8k394161
$endgroup$
An infinite union of countable sets may not be countable. For example, take the sets
$$A_x = {n+x| ninmathbb N}$$
so each set $A_x$ is countable, but $$bigcup_{xin[0,1]} A_x = mathbb R,$$
which is not countable.
However, your "square proof" would work fine to prove the statement
A countable union of countable sets is countable.
The proof would work because you can map the set $mathbb N$ to the countable union of countable sets, by mapping $1$ to the first element of the first set, then $2$ to the first element of the second set, $3$ to the first element of the first set, $4$ to the first element of the third set, and so on.
This statement would be very hard to prove by induction, however, because induction can only ever prove statements that are true for each element of $mathbb N$, not statements about the set as a whole.
answered Jun 14 '16 at 8:26
5xum5xum
91.8k394161
edited Jun 14 '16 at 8:31
answered Jun 14 '16 at 8:26
5xum5xum
91.8k394161
answered Jun 14 '16 at 8:26
5xum5xum
91.8k394161
answered Jun 14 '16 at 8:26
5xum5xum
91.8k394161
91.8k394161
$begingroup$
Awesome! Thanks alot. My textbook must have a type because it claims that an infinite union of countable sets is countable.
$endgroup$
– benssiao
Jun 14 '16 at 8:36
$begingroup$
@benssiao What book is that, and what page is the typo on, so we can all correct it in our copies?
$endgroup$
– bof
Jun 14 '16 at 8:52
$begingroup$
@benssiao Are you sure the textbook doesn't say "countably infinite"?
$endgroup$
– 5xum
Jun 14 '16 at 8:58
$begingroup$
Isnt countable and countably infinite the same? In the sense they are the same as saying a function is a bijection between N and A where A is any set. I have Stephen Abbott's second edition of Understanding Analysis. I am looking at theorem 1.5.8 ii on page 29.
$endgroup$
– benssiao
Jun 14 '16 at 9:06
$begingroup$
@benssiao Not entirely. Countable may also be finite. But even if you mean countable as "countably infinite", still, the terms infinite and countably infinite are certainly not the same.
$endgroup$
– 5xum
Jun 14 '16 at 9:09
|
show 2 more comments
$begingroup$
Awesome! Thanks alot. My textbook must have a type because it claims that an infinite union of countable sets is countable.
$endgroup$
– benssiao
Jun 14 '16 at 8:36
$begingroup$
@benssiao What book is that, and what page is the typo on, so we can all correct it in our copies?
$endgroup$
– bof
Jun 14 '16 at 8:52
$begingroup$
@benssiao Are you sure the textbook doesn't say "countably infinite"?
$endgroup$
– 5xum
Jun 14 '16 at 8:58
$begingroup$
Isnt countable and countably infinite the same? In the sense they are the same as saying a function is a bijection between N and A where A is any set. I have Stephen Abbott's second edition of Understanding Analysis. I am looking at theorem 1.5.8 ii on page 29.
$endgroup$
– benssiao
Jun 14 '16 at 9:06
$begingroup$
@benssiao Not entirely. Countable may also be finite. But even if you mean countable as "countably infinite", still, the terms infinite and countably infinite are certainly not the same.
$endgroup$
– 5xum
Jun 14 '16 at 9:09
$begingroup$
Awesome! Thanks alot. My textbook must have a type because it claims that an infinite union of countable sets is countable.
$endgroup$
– benssiao
Jun 14 '16 at 8:36
$begingroup$
Awesome! Thanks alot. My textbook must have a type because it claims that an infinite union of countable sets is countable.
$endgroup$
– benssiao
Jun 14 '16 at 8:36
$begingroup$
@benssiao What book is that, and what page is the typo on, so we can all correct it in our copies?
$endgroup$
– bof
Jun 14 '16 at 8:52
$begingroup$
@benssiao What book is that, and what page is the typo on, so we can all correct it in our copies?
$endgroup$
– bof
Jun 14 '16 at 8:52
$begingroup$
@benssiao Are you sure the textbook doesn't say "countably infinite"?
$endgroup$
– 5xum
Jun 14 '16 at 8:58
$begingroup$
@benssiao Are you sure the textbook doesn't say "countably infinite"?
$endgroup$
– 5xum
Jun 14 '16 at 8:58
$begingroup$
Isnt countable and countably infinite the same? In the sense they are the same as saying a function is a bijection between N and A where A is any set. I have Stephen Abbott's second edition of Understanding Analysis. I am looking at theorem 1.5.8 ii on page 29.
$endgroup$
– benssiao
Jun 14 '16 at 9:06
$begingroup$
Isnt countable and countably infinite the same? In the sense they are the same as saying a function is a bijection between N and A where A is any set. I have Stephen Abbott's second edition of Understanding Analysis. I am looking at theorem 1.5.8 ii on page 29.
$endgroup$
– benssiao
Jun 14 '16 at 9:06
$begingroup$
@benssiao Not entirely. Countable may also be finite. But even if you mean countable as "countably infinite", still, the terms infinite and countably infinite are certainly not the same.
$endgroup$
– 5xum
Jun 14 '16 at 9:09
$begingroup$
@benssiao Not entirely. Countable may also be finite. But even if you mean countable as "countably infinite", still, the terms infinite and countably infinite are certainly not the same.
$endgroup$
– 5xum
Jun 14 '16 at 9:09
|
show 2 more comments
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