# Cartesian Product in set theory

• Feb 8th 2010, 06:21 AM
pinco1984
Cartesian Product in set theory
Hi Folks,

I have came across a need and I am not aware if there is such a thing in set theory and if so what is it called.

Mainly I have several sets that I am interested in their cartesian product. But this cartesian product should not be a set of ordered pairs but a set of sets. Basically unordered pairs.

I wonder if this concept is well defined and what is it called.

Thanks.
P.
• Feb 8th 2010, 06:34 AM
wgunther
You mean something like:

$\displaystyle \{ \{x_1,x_2,\ldots\} \mid x_i\in X_i\}$

I can't think of any neat way to denote this. There's of course the idea of unordered n-tuples from a set A, which is denoted

$\displaystyle [A]^n$

But that's different than what you stated as it's n-tuples from 1 set, and not from a collection.
• Feb 8th 2010, 06:38 AM
pinco1984
The notation you have used is exactly what I need. But will it be true to call it as Cartesian Product?
• Feb 8th 2010, 06:52 AM
wgunther
No. Cartesian products are ordered. If you have say

$\displaystyle [A]^n$

This is also not a cartesian product. Cartesian products are an ordered n-tuple. The family of unordered n-tuples I don't think has any interesting name, except the set of unordered n-tuples! I never ran accross a reason to have to talk about them the way you did, as from a family of sets. The set of unordered n-tuples from a set is useful many areas, mostly when you talk about colorings (because normally you want to color say pairs of vertices (ie. an edge) rather than a binary relation on vertices (ie. an arc))
• Feb 8th 2010, 07:09 AM
pinco1984
Indeed not the unordered tuples of a single set but many of them is needed.

For the coloring problem I think what we need is the set of all sets of size 2, which is a memebr of the powerset of the set of vertexes.