Cartesian product

[kmath]

Hello everyone,

 

I want to make sure I understand the Cartesian product of two sets.

 

Let A = {1, 2}

B= {3, 4}

 

Then A X B = {(1,3),(1,4),(2,3),(2,4)}

 

Is that correct?

 

Thanks

[Nehushtan]

Correct.

[Amaton]

I haven't formally studied the context of this yet, but I have some questions...

 

1. If I have a set [math]S[/math], does the superscript [math]S^n[/math] denote a Cartesian product so many times (e.g. [math]\mathbb{R}^2[/math])? Or is this something else?

2. Can't I think of the Cartesian plane as basically a graphical representation of the Cartesian product of [math]\mathbb{R}[/math] and itself?

Edited April 20, 2013 by Amaton

[John]

1. That notation usually refers to the Cartesian product, yes.

2. Yes.

[Amaton]

1. That notation usually refers to the Cartesian product, yes.

2. Yes.

 

Thanks a lot. I'm inclined to continue...

 

1. What are other common uses of superscripts on sets then?

 

2. For [math]S^n[/math], is it okay to think of [math]n[/math] as representing spatial dimensions? Or is it more subtle and/or complex than this?

[John]

1. I'm not sure there are other *common* uses, but really you can use the same notation to describe just about anything you want (though for at least some definitions, other people might look at you funny). Wikipedia gives an example of using the notation for sums of vector spaces. Keep in mind I'm a student, not a professional mathematician. There may be very common uses of the notation that I haven't seen yet.

 

2. I don't see why not. For instance, [math]\mathbb{R}^n[/math] is commonly used to denote an n-dimensional Euclidean space, [math]\mathbb{C}^n[/math] for complex coordinate spaces, etc.