I suspect that this is a trivial thing, but I can't seem to find an example.

Say I have some Set indexed by Set $I$:

$$\{ x_i \}_{i \in I}$$

This called a "Family" of Sets, correct?

Next, say I want to use two indexing sets:

where $I$ and $J$ are Indexing Sets:

$$\{ x_{(i,j)} \}_{(i,j) \in (I \times J) } $$

Does this make sense to do? Is my notation correct? Thanks!

  • 1
    $\begingroup$ It makes as much sense as the first case because $I\times J$ is a set in its own right and because its elements are ordered pairs. $\endgroup$ – Git Gud Mar 21 '13 at 18:17

Yes, a set of sets $\{X_i\}$ indexed by some index set $I$ is indeed called a family of sets.

Your notation is correct, and emphasises that each $X_{(i,j)}$ has an ordered pair $(i,j)$ as an index, but conceptually all you need to know is that $I \times J$ is your index set, so you could also write it as $\{X_{u}\}_{u \in I \times J}$. However, your notation is also correct and probably preferable for clarity.

  • $\begingroup$ Great - this is very helpful. Thanks! $\endgroup$ – kevinpeterson Mar 21 '13 at 20:26

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