Is coutnable unions of finite sets an infinite set?

[MrGandalf]

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 $$\mathbb{N}$$?

 

[CRGreathouse]

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

Right. What's the problem?

 

[Cexy]

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.

 

[MrGandalf]

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.

 

[blue_raver22]

Hi there!

Yes, the resulting set will indeed be an infinite set. This is because the union of countably many finite sets is always infinite. In this case, since each set S_n only contains one element, the union S will be equal to the set of natural numbers, \mathbb{N}. This is because \mathbb{N} is defined as the set of all positive integers, including 1, 2, 3, and so on. Therefore, taking the countable union of all these sets will result in an infinite set. I hope this helps clarify the concept for you. Keep up the good work!