Understanding the Countability of Set S: 0s and 1s Infinite Union

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In summary, the equality ##S = \bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i \equiv \left\{{0,1}\right\}^\mathbb{N}## is false and the set on the right is uncountable while the set on the left is countable. This is because every element in ##S## is a finite tuple while every element in ##\left\{{0,1}\right\}^\mathbb{N}## is an infinite sequence. The confusion may have come from misunderstanding the set ##\left\{{0,1}\right\}^\mathbb{N}##, which has the power
  • #1
Bachelier
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##S = \bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i ##

methinks yes because:##S = \bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i \equiv \left\{{0,1}\right\}^\mathbb{N}##
 
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  • #2
Bachelier said:
##\bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i \equiv \left\{{0,1}\right\}^\mathbb{N}##

This equality is false. Furthermore, the set on the right is uncountable. The set on the left is countable.
 
  • #3
micromass said:
This equality is false. Furthermore, the set on the right is uncountable. The set on the left is countable.

So how do we look at ##\left\{{0,1}\right\}^∞##?
 
  • #4
Bachelier said:
So how do we look at ##\left\{{0,1}\right\}^∞##?

What do you mean with [itex]\infty[/itex]? The notation you are using now is not standard at all.
 
  • #5
Bachelier said:
##S = \bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i \equiv \left\{{0,1}\right\}^\mathbb{N}##
As micromass said, this is false. The reason for this is that every element of the left hand side is an n-tuple for some n, i.e., a FINITE tuple such as (0, 1, 0, 1, 1, 0). On the other hand, every element of the right hand side is an infinite sequence, such as (0, 1, 0, 1, 1, 0, ...). Therefore the left hand side and right hand side actually contain no elements in common.
 
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  • #6
I think that [itex]S = \bigcup _{i=1}^{∞}\left\{{0,1}\right\}^i[/itex] is countable all right. The mapping with [itex]\mathbb{N}[/itex] is quite obvious.
 
  • #7
Thanks guys, yes it is kind of clear that ## \bigcup_{i=1}^{∞}\left\{{0,1}\right\}^i ## is countable...I was just looking too much into it.

I believe my confusion was coming from misunderstanding the set: ##\left\{{0,1}\right\}^\mathbb{N}## which has the cardinality of the power set of ##\mathbb{N}##.
 

1. What does it mean for a set to be uncountable?

Uncountable sets are infinite sets that cannot be put into a one-to-one correspondence with the set of natural numbers. This means that there is no way to count or list all of the elements in an uncountable set.

2. How can you determine if a set is uncountable?

There are a few different ways to determine if a set is uncountable. One method is to use Cantor's diagonal argument, which states that if a set cannot be listed in a countable manner, then it must be uncountable. Another method is to show that a set is larger than the set of natural numbers, such as by showing that it has an uncountable number of elements.

3. Are all infinite sets uncountable?

No, not all infinite sets are uncountable. For example, the set of natural numbers is infinite but countable, as it can be listed in a one-to-one correspondence with itself. Only certain infinite sets, such as the set of real numbers, are uncountable.

4. Can an uncountable set have a countable subset?

Yes, it is possible for an uncountable set to have a countable subset. For example, the set of real numbers (which is uncountable) contains countable subsets such as the set of integers and the set of rational numbers.

5. What is the significance of uncountable sets in mathematics?

Uncountable sets are important in mathematics because they allow for the study of infinite sets that cannot be counted or listed in a straightforward manner. They also have applications in various branches of mathematics, such as in the study of real analysis and measure theory.

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