https://i.stack.imgur.com/zJVOs.jpg
This is what was posted. Can anybody help me understand why the function is defined to be f(i,j) = n(i-1)+j and not ij? I must be confused on what it's asking or cartesian products in general. There really isn't too much information on cartesian products in my book. Any help would be great.
Let's do it for e.g. $m=3$ and $n=4$
Constructing a bijection $\left\{ 1,2,3\right\} \times\left\{ 1,2,3,4\right\} \to\left\{ 1,2,\dots,11,12\right\} $ comes to the same as filling in the blanks in table:
$\begin{array}{c|ccccc} & 1 & 2 & 3 & 4\\\hline 1 \\ 2 \\ 3 \end{array}$
With numbers $1,2,\dots,12$.
A very natural way to do that is:
$\begin{array}{c|ccccc} & 1 & 2 & 3 & 4\\\hline 1 & 1 & 2 & 3 & 4\\ 2 & 5 & 6 & 7 & 8\\ 3 & 9 & 10 & 11 & 12 \end{array}$
If we search for a prescription of $f$ then we find: $$f\left(i,j\right)=4\left(i-1\right)+j$$