Intersection Distributes over Union/Family of Sets/Corollary
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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 of Family | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta} }\) | \(=\) | \(\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}\) | 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 of Family | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \bigcup_{\paren {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta}\) | \(=\) | \(\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}\) |
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families