Cartesian Product of Unions/General Result

Theorem

Let $I$ and $J$ be indexing sets.

Let $\left\langle{A_i}\right\rangle_{i \mathop \in I}$ and $\left\langle{B_j}\right\rangle_{j \mathop \in J}$ be families of sets indexed by $I$ and $J$ respectively.

Then:

$\displaystyle \left({\bigcup_{i \mathop \in I} A_i}\right) \times \left({\bigcup_{j \mathop \in J} B_j}\right) = \bigcup_{\left({i, j}\right) \mathop \in I \times J} \left({A_i \times B_j}\right)$

where:

$\displaystyle \bigcup_{i \mathop \in I} A_i$ denotes the union of $\left\langle{A_i}\right\rangle_{i \mathop \in I}$ and so on

$\times$ denotes Cartesian product.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families