Is coutnable unions of finite sets an infinite set?

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SUMMARY

The discussion centers on the properties of countable unions of finite sets within the context of natural numbers, denoted as \mathbb{N}. It is established that while finite unions of finite sets yield finite results, countable unions of finite sets result in countable sets. Specifically, the countable union of all natural numbers, represented as S = \bigcup_{i\in\mathbb{N}}S_i, is indeed infinite and equivalent to \mathbb{N}. The participants confirm these mathematical principles, clarifying any confusion regarding the nature of these unions.

PREREQUISITES
  • Understanding of set theory concepts, particularly unions and cardinality.
  • Familiarity with natural numbers and their notation, \mathbb{N}.
  • Basic knowledge of finite and countable sets.
  • Ability to interpret mathematical notation and expressions.
NEXT STEPS
  • Study the properties of finite sets and their unions.
  • Explore the concept of countable sets in more depth.
  • Learn about the implications of the Axiom of Countable Choice in set theory.
  • Investigate the differences between finite, countable, and uncountable sets.
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Mathematicians, students of mathematics, and anyone interested in set theory and its applications will benefit from this discussion.

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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