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

In summary, the image below represents the Cartesian product of the set of natural numbers with itself, denoted by N^2. This means the set of all ordered pairs of natural numbers. To represent non-pair numbers, one can use the notation ∀k,l,m∈N×N×N | k>1, l>2, m>k+l or ∀k,l,m∈N | k>1, l>2, m>k+l. The former represents an ordered triplet while the latter represents three individual numbers.
  • #1
MienTommy
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0
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?
 

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  • #2
MienTommy said:
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
$$\forall_{ \begin{array} ((k,l,m) \in \mathbb{N}\times\mathbb{N}\times\mathbb{N} \\ k > 1 \\ l > 2 \\ m>k+l \end{array}}$$
 
  • #3
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?
 
Last edited:
  • #4
MienTommy said:
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.
 
  • #5
I see. What is the difference between the ordered triplet and the other?
 
  • #6
MienTommy said:
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.
 
  • #7
MienTommy said:
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.
 

1. What is the meaning of N^2 in the case of natural numbers?

N^2 in the case of natural numbers refers to the square of a natural number, which is obtained by multiplying the number by itself. For example, N^2 of 3 would be 9, as 3 multiplied by itself equals 9.

2. How is N^2 represented in mathematical notation?

In mathematical notation, N^2 is represented as N², where the small 2 above N indicates the exponent or power to which N is raised.

3. What is the difference between N^2 and 2N in the case of natural numbers?

N^2 represents the square of a natural number, while 2N represents twice the value of a natural number. For example, N^2 of 4 is 16, while 2N of 4 is 8.

4. What is the relationship between N^2 and N?

The square of a natural number N is equal to the product of N and itself. In other words, N^2 is the result of multiplying N by N.

5. How is N^2 used in scientific calculations?

N^2 is used in various scientific calculations, such as calculating areas and volumes. For example, the area of a square with side length N is given by N^2, and the volume of a cube with side length N is given by N^3.

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