User:Dr Who/Sandbox/array

Working since last summer in my spare time on a rigorous definition of array and matrix which ground on sets, ordered pairs and tuples only, without any reference to maps.

1. Introduce the concept of ordered pairs and tuples whose elements can be ordered pairs and tuples as well;

2. Show that the row of an array can be defined as an ordered pair whose 1st element is the corresponding tuple, and 2nd element has card=1

3. Show that the column of an array can be defined as an ordered pair whose 1st element has card=1, and 2nd element is the corresponding tuple.

.....

4. In the end of the process, I will show how to define (or "build") an array and write down it as an ordered pair, where the first element is the tuple of rows, and the second element is the tuple of columns (provided that a certain requirement is satisfied); one of the nice side effects is that we invert the order of the two elements of the array's ordered pair, we have a transpose array, hence a transpose matrix.


 * A bit like:
 * $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

expressed as:
 * $\left({\left({1, 2}\right), \left({3, 4}\right)}\right)$

...or something more subtle?

This is the way the existence of arrays / matrices is justified in the context of axiomatic set theory. --prime mover 17:07, 31 December 2011 (CST)


 * Well, yes, what I have in mind is a bit more complicated than that--Dr Who 19:25, 31 December 2011 (CST)


 * Can you express it symbolically? I'm missing something. --prime mover 02:45, 1 January 2012 (CST)