Cartesian product of index family of sets

[chipotleaway]

Cartesian product of indexed family of sets

The definition of a Cartesian product of an indexed family of sets $(X_i)_{i\in I}$ is $\Pi_{i\in I}X_i=\left\{f:I \rightarrow \bigcup_{i \in I} \right\}$

So if I understand correctly, it's a function that maps every index $i$ to an element $f(i)$ such that $f(i) \in X_i$…my question is, is there supposed to be a notion of 'order' implied in the definition here?

The 'usual' Cartesian product is $\Pi^n_{i=1}X_i=\left\{ (x_i)^n_{i=1}: x_i \in X_i \right\}$ and here, there seems to be some notion of 'order' cause the first element is in $X_1$, second in $X_2$ etc.

But for the first, if the index set is I={1,2,3}, then couldn't we have have $X_1 \times X_2 \times X_3$ or $X_2\times X_1\times X_3$ (any reordering of indices in the index set)? Then the definition of the Cartesian product could be $((f(1), f(2), f(3))$ or $(f(2), f(1), f(3))$...

 

[tiny-tim]

hi chipotleaway!

… is there supposed to be a notion of 'order' implied in the definition here?

yes and no

there's only an order if the index set, I, has an order

 

[chipotleaway]

hmm...so the case if I={1,2,3,4}, it would only reduce to the 'usual definition', n running from 1 to 4 to we somehow define the order to be 1,2,3,4

 

[tiny-tim]

yes, eg spacetime coordinates are sometimes written (x₀, x₁, x₂, x₃), and sometimes (x₁, x₂, x₃, x₄)

ie sometimes with t (or ct) at the beginning and sometimes with t at the end

 

[blue_raver22]

am I on the right track here?

Yes, you are correct. The Cartesian product of an indexed family of sets does not have a specific order implied in its definition. The order in which the elements of the product are written is arbitrary and can vary depending on the chosen index set. For example, if the index set is {1, 2, 3}, then the Cartesian product could be written as X_1 \times X_2 \times X_3 or X_2 \times X_1 \times X_3, as you mentioned. Both representations are valid and equivalent.

The key difference between the usual Cartesian product and the Cartesian product of an indexed family of sets is that the latter allows for a more general structure, where the index set can be infinite and the sets in the family can vary in size and type. This allows for a more flexible and versatile way of representing mathematical objects and relationships.

In summary, the Cartesian product of an indexed family of sets is a mathematical concept that does not have a specific order implied, and the order of the elements can vary depending on the chosen index set. Its purpose is to provide a general and flexible way of representing mathematical objects and relationships.