# Is coutnable unions of finite sets an infinite set?

• MrGandalf
In summary, the conversation discusses the result of taking the countable union of all natural numbers in a set. It is clarified that the resulting set will be infinite and equal to the set of natural numbers, and that unions of finite sets are finite while unions of countable sets are countable. The conversation ends with appreciation for clearing up any confusion and an apology for a typo in the thread title.
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 $$\mathbb{N}$$?

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.

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!

## 1. Is it possible for the countable union of finite sets to be an infinite set?

Yes, it is possible for the countable union of finite sets to be an infinite set. This is because the union of finite sets can result in an infinite set if the individual sets have an infinite number of elements or if the sets overlap in elements.

## 2. What is a countable union of finite sets?

A countable union of finite sets is a mathematical concept where a collection of finite sets is combined into one set by taking the elements from all the individual sets and putting them together. This union is countable, meaning that the elements in the set can be counted and listed in a specific order.

## 3. How does the countable union of finite sets differ from the union of infinite sets?

The main difference between the countable union of finite sets and the union of infinite sets is that the countable union of finite sets always results in a countable set, while the union of infinite sets can result in a countable or an uncountable set depending on the specific sets being combined.

## 4. What is an example of a countable union of finite sets that results in an infinite set?

An example of a countable union of finite sets that results in an infinite set is the union of the sets of even and odd numbers. Both sets are finite, but when combined, they result in an infinite set of all integers.

## 5. How is the concept of countable union of finite sets relevant in mathematics?

The concept of countable union of finite sets is relevant in many areas of mathematics, including set theory, topology, and number theory. It allows for the creation of larger sets from smaller ones and is an important tool in proof techniques and constructions in various mathematical fields.

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