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

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The set {m+n | m,n ∈ Z} is proven to be countable by establishing a one-to-one correspondence with the integers. A proposed method involves creating an NxN scheme, where integers are arranged along the sides of a grid, allowing for diagonal counting to encompass all elements of the set. This approach, while valid, may be unnecessarily complex, as the set can be directly equated to Z, simplifying the proof process.

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Homework Statement


Prove that {m+n, m,n [itex]\in[/itex]Z} is countable


Homework Equations





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?
 
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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 [itex]\in[/itex]Z} is countable


Homework Equations





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.
 

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