Mathematical notation to represent all combinations of three variables?

Question

This is a fairly simple question involving mathematical notation which is outside of my expertise. I'm looking for the correct representation of a matrix M which shows all combinations of three variables, i, j & k. This is coded up in MATLAB as something like:

M = [];
for i = 0:5:20
    for j = 0:2:10
        for k = 0:3:12
            M(end+1,:) = [i,j,k];
        end
    end
end

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This produces a 150x3 matrix for M, here is what the first 20 lines of M look like:

 0     0     0
 0     0     3
 0     0     6
 0     0     9
 0     0    12
 0     2     0
 0     2     3
 0     2     6
 0     2     9
 0     2    12
 0     4     0
 0     4     3
 0     4     6
 0     4     9
 0     4    12
 0     6     0
 0     6     3
 0     6     6
 0     6     9
 0     6    12

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So how to represent M mathematically? The first thing that comes to mind is something like this:

\begin{equation} \sum\limits_{i=0}^4 \sum\limits_{j=0}^5 \sum\limits_{j=0}^4 5i . 2j . 3k \end{equation}

But obviously this represents the sum of the elements in M, not the matrix M itself. I'm thinking the correct representation will involve the "$\forall$" operator somehow, eg. $\forall$ ..... $\forall$ ..... $\forall$ .....

Answer 1

Let your variable $i$ belong to some set $\mathbb{I}$, $j$ to $\mathbb{J}$, and $k$ to $\mathbb{K}$. Then the set of all combinations can be represented by $\mathbb{I} \times \mathbb{J} \times \mathbb{K}$.

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By the way, here's a two-line solution to your MATLAB code that avoids looping. (Never loop in MATLAB if you can avoid it).

I = 0:5:20; J = 0:2:10; K = 0:3:12;
M = [kron(I',ones(length(J)*length(K),1)) kron(repmat(J',length(I),1),ones(length(K),1)) repmat(K',length(I)*length(J),1)]