Difference between theorems 2.12 & 214 of Baby Rudin

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Please help me in understanding the difference between theorems 2.12 & 2.14 of Rudin's Principles of Mathematical Analysis.

Both are sets of sequences.
Set S in Th.2.12 is union of countable sequences
While set A in Th 2.14 is set of "all" sequences.

Is set A uncountable only because it has "all" sequences, whereas set A is countable because it does not have "all" sequences, but only countable sequences?

Thanks in advance.
 
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Answers and Replies

  • #2
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Please help me in understanding the difference between theorems 2.12 & 2.14 of Rudin's Principles of Mathematical Analysis.

Both are sets of sequences.
$${}$$

No, they aren't. In theorem 2.12 Rudin proves the important theorem that a countable union of countable sets is countable, whereas in theorem 2.14 he proves that the set of all sequences formed only with 0's and 1's is uncountable.

DonAntonio

Set S in Th.2.12 is union of countable sequences
While set A in Th 2.14 is set of "all" sequences.

Is set A uncountable only because it has "all" sequences, whereas set A is countable because it does not have "all" sequences, but only countable sequences?

Thanks in advance.
 
  • #3
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Thanks for the reply DonAntonio. You say

whereas in theorem 2.14 he proves that the set of all sequences formed only with 0's and 1's is uncountable.

So,since th. 2.14 is true for 1's & 0's, it has to be true for any natural/rational numbers.
And set of all sequences is Union.

If th 2.12 says a countable union of countable sets is countable.
so can i say that th 2.14 states union of all countable sets is uncountable?

Anything wrong?:confused:
 
  • #4
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Thanks for the reply DonAntonio. You say




So,since th. 2.14 is true for 1's & 0's, it has to be true for any natural/rational numbers.
And set of all sequences is Union.

If th 2.12 says a countable union of countable sets is countable.
so can i say that th 2.14 states union of all countable sets is uncountable?

Anything wrong?:confused:

"All countable sets"? I'm afraid this might be a being that cannot be a set, but anyway: in your first

line, do you mean to ask whether the set of all (real) sequences is uncountable? The answer is yes to this question.

I've no idea what you mean by "And the set of all sequences is Union".

DonAntonio
 
  • #5
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If th 2.12 says a countable union of countable sets is countable.
so can i say that th 2.14 states union of all countable sets is uncountable?
Surely you can see that even if a countable union of countable sets is countable; that doesn't mean that an arbitrary union of countable sets is countable.

Do you understand what a countable union of countable sets means? It means you have a countable set, and another countable set, and another countable set ... and altogether you have a countable collection of countable sets. The union of all those sets must be countable.

One issue that may be in play here is that you are not supposed to be learning set theory from Rudin. Rudin is simply reviewing the basics of set theory so that he can go on to teach real analysis. A decent familiarity with the basics of set theory is a prerequisite for a course in real analysis. If this is your first exposure to this material, you may need to work extra hard for a while and have a chat with your instructor about this.
 
  • #6
Bacle2
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To put it a bit differently , in 2.12 , the result is that :

2.12: A countable union of countable sets is countable. Notice that the sequences

are indexed by the natural numbers --notice that the sequences are indexed

by n, i.e., we have the sequences A1, A2,..........


2.14: There is a reference to all the sequences formed by 0's and 1's. But, from a

"known" result, the set of all sequences consisting only of 0's and 1's, is not

countable. do you know what this "known: result is ?
 
  • #7
Bacle2
Science Advisor
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Thanks for the reply DonAntonio. You say




So,since th. 2.14 is true for 1's & 0's, it has to be true for any natural/rational numbers.
And set of all sequences is Union.

If th 2.12 says a countable union of countable sets is countable.
so can i say that th 2.14 states union of all countable sets is uncountable?

Anything wrong?:confused:
No; in 2.12, the collection of sequences you're given is countable, i.e., you're told that

you have countably-many sequences, each sequence itself being --by definition--countable.

Rudin shows that the union of these is countable.

In 2.14, you are given a collection of sequences, but, in this case, the collection of

sequences is not a countable, i.e., there are more than countably-many sequences that

contain only 0's and 1's.
 
  • #8
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Thanks everyone for replying. I will refer some other books & try to dig some more.

In meantime I will try to put my question in other words.

In Pg 26 he says "more loosely, we may say that elements of any countable set can be "arranged in a sequence""

So I am assuming that, if anything is arranged in sequence then it is countable.

Now, in th 2.14, Rudin is constructing a sequence of 0's & 1's...

Lats say we have only 2 sets of sequence
0101000...
1001000...

Then the union of these is 2 sequences is countable. (Th 2.12)

Likewise if I have countable sets of such sequences, then its union should be countable(or finite as the case may be).
( By assigning 0's & 1's to x11, x12, etc,.. in th 2.12 )

But in th 2.14 the union is not countable.
This is possible if the collection of sets he is taking is more than countable (i.e uncountable)

So is my understanding correct?
Or is it right time for me to look into some other books? :smile:
 
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