Cartesian product of omega tuples

[EV33]

Homework Statement

So I am just trying to understand the concepts here.

My main question is what exactly is the cartesian product of an omega tuple?

Homework Equations

Given a set X, we define an ω-tuple of elements of X to be a function
x:N$\rightarrow$X

We denote x as
Let {A1,A2,...} be a family of sets

Let X be the union of of the sets in this family.


The book claims that the cartesian product of this indexed family of sets is denoted A1XA2...
and is defined to be the set of all ω-tuples (x1,x2,...) of elements of X such that xi is in Ai for each i, which is denoted by X^ω.

The Attempt at a Solution

So is X^ω=x? because that is what the book makes it sound like to me.

Also, the book tells me what the cartesian product of the indexed family is but it doesn't explain what the cartesian product of an omega tuple is. Can anyone explain what it is to me.

 

[mighty2000]

it is important to have a clear understanding of the concepts you are working with. The cartesian product of an omega tuple is essentially the set of all possible combinations of elements from each set in the indexed family. So in this case, Xω would be the set of all possible ω-tuples (x1,x2,...) where xi is an element of Ai for each i. It is denoted by Xω because it is the set of all possible ω-tuples from the family of sets X. It is not the same as x, as x is just one specific ω-tuple while Xω is the set of all possible ω-tuples. I hope this helps clarify the concept for you. If you have any further questions, please don't hesitate to ask.