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