Union Distributes over Intersection/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:

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

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

Proof

\(\ds \bigcap_{\alpha \mathop \in I} \paren {A_\alpha \cup B}\)

\(=\)

\(\ds \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B\)

Union Distributes over Intersection of Family

\(\ds \leadsto \ \ \)

\(\ds \bigcap_{\alpha \mathop \in I} \paren {A_\alpha \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta} }\)

\(=\)

\(\ds \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}\)

setting $\ds B = \paren {\bigcap_{\beta \mathop \in J} B_\beta}$

\(\ds \leadsto \ \ \)

\(\ds \bigcap_{\alpha \mathop \in I} \paren {\bigcap_{\beta \mathop \in J} \paren {A_\alpha \cup B_\beta} }\)

\(=\)

\(\ds \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}\)

Union Distributes over Intersection of Family

\(\ds \leadsto \ \ \)

\(\ds \bigcap_{\paren {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cup B_\beta}\)

\(=\)

\(\ds \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}\)

$\blacksquare$

Sources

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