# Proving Countability of {m+n, m,n \inZ} Using a NxN Scheme

• aaaa202
In summary: It's a bit of a confusing notation since the set is defined in terms of elements from Z itself, but I think it's just a different way of saying Z.
aaaa202

## Homework Statement

Prove that {m+n, m,n $\in$Z} is countable

## The Attempt at a Solution

I Can prove it if I make a nxn scheme and put 1,-1,2,-2 along each side. This generates a table which when counted a long first,second etc. Diagonal hits all the numsers in the given set. But is this the formal Way to prove these kinds of things?

Isn't that set just equal to Z again? Maybe I'm just misunderstanding notation...

aaaa202 said:

## Homework Statement

Prove that {m+n, m,n $\in$Z} is countable

## The Attempt at a Solution

I Can prove it if I make a nxn scheme and put 1,-1,2,-2 along each side. This generates a table which when counted a long first,second etc. Diagonal hits all the numsers in the given set. But is this the formal Way to prove these kinds of things?
The set could also be described as {p | p = m + n, where m, n ##\in## Z}. All you need to do is to establish a one-one pairing with the integers. The things in the set are just numbers, not ordered pairs, so based on the notation you've used, your table is way more complicated than what is needed.

johnqwertyful said:
Isn't that set just equal to Z again? Maybe I'm just misunderstanding notation...
That's how I read it as well.

## 1. What does it mean for a set to be countable?

A countable set is a set that can be placed into a one-to-one correspondence with the set of natural numbers (1, 2, 3, ...). This means that each element in the set can be paired with a unique natural number, and there will always be a finite or infinite number of elements in the set.

## 2. How do you prove that a set is countable?

The most common method to prove that a set is countable is by constructing a bijection (a one-to-one and onto function) between the set and the set of natural numbers. This can be done by listing out the elements in a systematic way, such as in a table, or by using a formula.

## 3. Can an infinite set be countable?

Yes, an infinite set can be countable. For example, the set of all positive even numbers (2, 4, 6, ...) is infinite but countable because it can be paired with the set of natural numbers by the function f(n) = 2n.

## 4. Are all subsets of a countable set also countable?

No, not all subsets of a countable set are countable. For example, the set of real numbers is uncountable, but it contains subsets such as the set of rational numbers which are countable.

## 5. Is the set of irrational numbers countable?

No, the set of irrational numbers is uncountable. This means that there is no bijection between the set of irrational numbers and the set of natural numbers, and thus it cannot be paired with a unique natural number for every element.

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