What does N^2 mean in the case of natural numbers?

[MienTommy]

What does the N^2 mean in this case? (Image below)

Does it mean, for all two pairs of natural numbers, a and b?

How would I represent non pair numbers, i.e. how would I write "For integers k,l, and m such that k>1, l>2, m>k+l" all in one line?

 

[fresh_42]

What does the N^2 mean in this case? (Image below)

It means the Cartesian product $\mathbb{N} \times \mathbb{N}$

Does it mean, for all two pairs of natural numbers, a and b?

Yes.

How would I represent non pair numbers, i.e. how would I write "For integers k,l, and m such that k>1, l>2, m>k+l" all in one line?

You just did, didn't you? What are non pair numbers? Or did you mean
$$\forall_{ \begin{array} ((k,l,m) \in \mathbb{N}\times\mathbb{N}\times\mathbb{N} \\ k > 1 \\ l > 2 \\ m>k+l \end{array}}$$

 

[MienTommy]

Yes that is what I meant. I wanted to know the symbol form.
Would you add a such that symbol '|' after N×N×N?

If I removed the parenthesis, ∀k,l,m∈N×N×N | k>1, l>2, m>k+l
Then would it mean the same thing?

 

[fresh_42]

Yes that is what I meant. I wanted to know the symbol form. Would you add a such that symbol '|' after N×N×N? If I removed the parenthesis, ∀k,l,m∈N×N×N | k>1, l>2, m>k+l

Then would it mean the same thing?

If you remove the parenthesis, then one $\mathbb{N}$ is enough. There is a subtle difference between them: $k,l,m \in \mathbb{N}$ are simply three natural numbers, whereas $(k,l,m) \in \mathbb{N}^3$ is a ordered triplet. In most cases this doesn't really matter, but rigorously it's not the same. And of course one wouldn't actually write all conditions below each other since it's impractical. An alternative would be to write $\forall_{k,l,m \in \mathbb{N}} \,\text{ with }\, k>1\,,\,l>2\,,\,m>k+l \;:\;$ etc.

 

[MienTommy]

I see. What is the difference between the ordered triplet and the other?

 

[fresh_42]

I see. What is the difference between the ordered triplet and the other?

$(1,2,3) \neq (2,3,1)$ but $1,2,3$ are only three numbers.

 

[Zafa Pi]

What does the N^2 mean in this case? (Image below)

Does it mean, for all two pairs of natural numbers, a and b?

How would I represent non pair numbers, i.e. how would I write "For integers k,l, and m such that k>1, l>2, m>k+l" all in one line?

If A and B are sets A^B is the set of all maps of B into A. In your case 2 stands for a set with two elements, {1,2} for example.