# Cartesian product of index family of sets

1. Mar 8, 2014

### 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 $(f(2), f(1), f(3))$...

Last edited: Mar 8, 2014
2. Mar 9, 2014

### tiny-tim

hi chipotleaway!
yes and no

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

3. Mar 9, 2014

### 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

4. Mar 9, 2014

### 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