Union Distributes over Intersection/Family of Sets/Corollary

From ProofWiki
Jump to navigation Jump to search

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:

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

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


Proof

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

$\blacksquare$


Sources