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

  1. Nov 12, 2016 #1
    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?

    Attached Files:

  2. jcsd
  3. Nov 12, 2016 #2


    Staff: Mentor

    It means the Cartesian product ##\mathbb{N} \times \mathbb{N}##
    You just did, didn't you? What are non pair numbers? Or did you mean
    $$\forall_{ \begin{array} ((k,l,m) \in \mathbb{N}\times\mathbb{N}\times\mathbb{N} \\ k > 1 \\ l > 2 \\ m>k+l \end{array}}$$
  4. Nov 12, 2016 #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: Nov 12, 2016
  5. Nov 12, 2016 #4


    Staff: Mentor

    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.
  6. Nov 12, 2016 #5
    I see. What is the difference between the ordered triplet and the other?
  7. Nov 12, 2016 #6


    Staff: Mentor

    ##(1,2,3) \neq (2,3,1)## but ##1,2,3## are only three numbers.
  8. Dec 14, 2016 #7

    Zafa Pi

    User Avatar
    Gold Member

    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.
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