Intersection Distributes over Union/Family of Sets/Corollary

Theorem

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

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be indexed families of subsets of a set $S$.

Then:

$\ds \bigcup_{\tuple {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}$

where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.

Proof

 $\ds \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B}$ $=$ $\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$ Intersection Distributes over Union: Family of Sets $\ds \leadsto \ \$ $\ds \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta} }$ $=$ $\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}$ setting $\ds B = \paren {\bigcup_{\beta \mathop \in J} B_\beta}$ $\ds \leadsto \ \$ $\ds \bigcup_{\alpha \mathop \in I} \paren {\bigcup_{\beta \mathop \in J} \paren {A_\alpha \cap B_\beta} }$ $=$ $\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}$ Intersection Distributes over Union: Family of Sets $\ds \leadsto \ \$ $\ds \bigcup_{\paren {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta}$ $=$ $\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}$

$\blacksquare$