# Measuring Sets: Finite vs. Countably Infinite

• zeebo17
In summary, finite sets have a limited number of elements while countably infinite sets have an infinite number of elements that can be counted and put into a one-to-one correspondence with the set of natural numbers. The way to determine if a set is finite or countably infinite is by counting its elements, and a set cannot be both finite and countably infinite. Real-life examples of countably infinite sets include the set of even numbers, prime numbers, and rational numbers. The concept of "size" is different for finite and countably infinite sets as the size of a finite set is determined by the number of elements it contains, while the size of a countably infinite set is determined by its ability to be put into a one-to-one correspondence
zeebo17
I just started learning some basic measure theory.

Could someone explain the difference between $$\overline{F(A \times A)}$$ and $$\overline{F(A) \times F(A)}$$ where A is a finite set. Also, how would this be different in A was an countably infinite set?

Thanks!

zeebo17 said:
where A is a finite set.

It is nice you told us what A is. But you didn't say what F and the bar are.

## 1. What is the difference between finite and countably infinite sets?

A finite set is a set that contains a specific, limited number of elements. A countably infinite set is a set that contains an infinite number of elements, but can be counted and put into a one-to-one correspondence with the set of natural numbers.

## 2. How do you determine if a set is finite or countably infinite?

A set is finite if its elements can be counted and there is a specific number of elements. A set is countably infinite if its elements can be counted and put into a one-to-one correspondence with the set of natural numbers.

## 3. Can a set be both finite and countably infinite?

No, a set cannot be both finite and countably infinite. A set is either finite or countably infinite, but not both.

## 4. Are there any real-life examples of countably infinite sets?

Yes, there are many real-life examples of countably infinite sets. Some examples include the set of all even numbers, the set of all prime numbers, and the set of all rational numbers.

## 5. How is the concept of "size" different for finite and countably infinite sets?

The size of a finite set is determined by the number of elements it contains. However, the size of a countably infinite set cannot be determined by counting its elements, as it has an infinite number of elements. Instead, the size of a countably infinite set is determined by its ability to be put into a one-to-one correspondence with the set of natural numbers.

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