Is coutnable unions of finite sets an infinite set?

AI Thread Summary
The discussion centers on the nature of countable unions of finite sets within the context of natural numbers. When taking the countable union of sets S_i, where each S_i contains a single natural number, the resulting set S is indeed infinite and equivalent to the set of natural numbers, \mathbb{N}. It is clarified that finite unions of finite sets remain finite, while countable unions of finite sets yield a countable set. Additionally, the properties of unions of countable sets are confirmed, reinforcing that countable unions of countable sets are also countable. The initial confusion is resolved, affirming the understanding of these set properties.
MrGandalf
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Hiya. :)

While doing an assignment I ran into this little problem.

We are working in the set of natural numbers \mathbb{N}.

If i collect each natural number in a set
S_1 = \{1\}, S_2 = \{2\},\ldots, S_n = \{n\},\ldots

What happens when I take the countable union of all these?
S = \bigcup_{i\in\mathbb{N}}S_i

The resulting set will be an infinite set, right? It will be equal to \mathbb{N}?
 
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MrGandalf said:
The resulting set will be an infinite set, right? It will be equal to \mathbb{N}?

Right. What's the problem?
 
Yup.

Finite unions of finite sets are finite.

Countable unions of finite sets are countable.

Finite unions of countable sets are countable.

Countable unions of countable sets are countable.
 
Thanks.

I was just really unsure there for a moment, but I think I see it now.
Thanks for clearing that up for me.

PS Sorry about the typo in the thread title.
 

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