Cartesian Product of Unions/General Result

Theorem

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

Let $\family {A_i}_{i \mathop \in I}$ and $\family {B_j}_{j \mathop \in J}$ be families of sets indexed by $I$ and $J$ respectively.

Then:

$\ds \paren {\bigcup_{i \mathop \in I} A_i} \times \paren {\bigcup_{j \mathop \in J} B_j} = \bigcup_{\tuple {i, j} \mathop \in I \times J} \paren {A_i \times B_j}$

where:

$\ds \bigcup_{i \mathop \in I} A_i$ denotes the union of $\family {A_i}_{i \mathop \in I}$ and so on

$\times$ denotes Cartesian product.

Proof

$\ds $

$\ds \tuple {x, y} \in \paren {\bigcup_{i \mathop \in I} A_i} \times \paren {\bigcup_{j \mathop \in J} B_j}$

$\ds \leadstoandfrom \ \ $

$\ds $

$\ds \paren {\exists i \in I: x \in A_i}$

$\ds $

$\land$

$\ds \paren {\exists j \in J: y \in B_j}$

Definition of Cartesian Product and Definition of Set Union

$\ds \leadstoandfrom \ \ $

$\ds $

$\ds \exists i \in I \exists j \in J: \paren {x \in A_i} \land \paren {y \in B_j}$

$\ds \leadstoandfrom \ \ $

$\ds $

$\ds \exists \tuple {i, j} \in I \times J: \tuple {x, y} \in A_i \times B_j$

Definition of Cartesian Product

$\ds \leadstoandfrom \ \ $

$\ds $

$\ds \tuple {x, y} \in \bigcup_{\tuple {i, j} \mathop \in I \times J} \paren {A_i \times B_j}$

Definition of Set Union

$\blacksquare$

Sources

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