An easy question about cartesian product

In summary: A->B->C and g: (AxB)->C are naturally isomIn summary, the conversation discusses the definition of a cartesian product of two sets A and B, as well as the notation for multiple cartesian products. The participants also touch on the concept of isomorphism in relation to vector spaces and functions, and conclude that the cartesian product is associative.
  • #1
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Hi everybody,
When we have two sets A and B , we define the cartesian product of A and B as the set A*B={(x,y): (x element of A) and (y element of B)}. We also define A*A*...*A (n factors)=A^n. So when we write (A^2)*B, this is the same as A*A*B? I mean, for example (R^2)*R is the same as R*R*R, or ((1,2),3)=(1,2,3) ?
Thanks
 
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  • #2
They're isomorphic as vector spaces, i.e. they are the same. (unless you want to get hyper formal, in which case I think you need to stick with just isomorphic).
 
  • #3
Hum, I answered for R as a vector space, sorry. Given any 2 sets, then you have a 1-1 and onto mapping between (A^2)xB and AxAxB.
 
  • #4
Thanks for your answer. Actually i am asking this question because I read somewhere that the graph of a function f (if it's called this way) is the set G={(x,f(x)):x element of f's domain}. So if f: (R^2) -> R then the vectors (x,y,f(x,y) that are points of f's 3D representation must be the same as ((x,y),f(x,y)). ( I hope u understood what i am asking).

Thanks again
 
  • #5
Yes, I think I do.
And yes, they are the same. Eventually, it's just 2 different ways of looking at it- just like thinking of f(x,y) as a function of 2 scalar variables or of 1 vector variable.
I hope you understand what I'm trying to say.:)
 
  • #6
To put in "proper" terms (ie ones that might help you search for other things on it) you're getting towards the idea that the cartesian product is associative: that is there are natural isomorphisms from (AxB)xC to Ax(BxC)
 

1. What is a cartesian product?

A cartesian product is a mathematical operation that combines two sets to form a new set. It is denoted by the symbol x or ∪ and is used to find all possible combinations of elements from both sets.

2. How is a cartesian product calculated?

A cartesian product is calculated by pairing each element from one set with every element from the other set. For example, if set A = {1,2} and set B = {a,b}, the cartesian product would be {1a, 1b, 2a, 2b}.

3. What is the difference between a cartesian product and a cross product?

A cartesian product and a cross product are two different operations in mathematics. While a cartesian product combines two sets, a cross product combines two vectors in three-dimensional space to form a new vector.

4. How is a cartesian product used in real life?

Cartesian products are commonly used in computer science and data analysis, particularly in databases. They can also be used to find all possible outcomes in probability and to create visual representations of data in two or more dimensions.

5. What is the inverse of a cartesian product?

The inverse of a cartesian product is the intersection of two sets. It is denoted by the symbol ∩ and represents all the elements that the two sets have in common.

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