Is coutnable unions of finite sets an infinite set?

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Discussion Overview

The discussion revolves around the properties of unions of finite sets within the context of natural numbers. Participants explore whether the countable union of finite sets results in an infinite set and its relationship to the set of natural numbers.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant presents a scenario involving the collection of natural numbers in finite sets and questions the nature of their countable union.
  • Another participant affirms that the resulting set from the countable union of these finite sets is indeed infinite and suggests it is equal to the set of natural numbers.
  • A third participant outlines several properties of unions, stating that finite unions of finite sets are finite, while countable unions of finite sets are countable.
  • This participant also notes that finite unions of countable sets and countable unions of countable sets are countable.

Areas of Agreement / Disagreement

Participants generally agree that the countable union of finite sets results in a countable set, but the discussion does not resolve whether this set is definitively equal to the set of natural numbers.

Contextual Notes

The discussion does not address potential limitations or assumptions regarding the definitions of countable and finite sets, nor does it explore the implications of these properties in different contexts.

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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