Re: less-than-uncool preschool
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Reply #17 –

Note that when a set is given by an explicit list of its elements, it doesn't matter what order those elements are listed in. {2, 3, 5, 7} and {5, 2, 7, 3} are exactly the same set.

But there are times when we are concerned with order. In particular, let's talk about "ordered pairs".

An "ordered pair" is an object consisting of two other objects, considered in a specific order. So if *a* and *b* are objects of some sort (perhaps numbers), then we write (*a*, *b*) for the ordered pair consisting of (first) *a* and (then) *b* (in that order). The objects within an ordered pair can be termed the "coordinates" of the pair.

The "Cartesian product" of two sets *A* and *B* is this:

*A* × *B* = {(*a*, *b*) | *a* ∈ *A* and *b* ∈ *B*}.

In other words, it's the set of all ordered pairs where the first coordinate is an element of *A* and the second coordinate is an element of *B*.

Example: If *A* = {2, 3, 5} and *B* = {1, 3, 5, 7}, then

*A* × *B* = {(2, 1), (2, 3), (2, 5), (2, 7), (3, 1), (3, 3), (3, 5), (3, 7), (5, 1), (5, 3), (5, 5), (5, 7)}

We could go beyond ordered pairs and consider "ordered triples" such as (*a*, *b*, *c*) or even "ordered *n*-tuples" such as (*x*_{1}, *x*_{2}, ..., *x*_{n}), but let's get some other groundwork out of the way before we go there.