Set Union is Self-Distributive/Families of Sets

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $I$ be an indexing set.

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


Then:

$\ds \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}$

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


Proof

\(\ds x\) \(\in\) \(\ds \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha}\)
\(\ds \leadsto \ \ \) \(\ds \exists \beta \in I: \, \) \(\ds x\) \(\in\) \(\ds A_\beta \cup B_\beta\) Definition of Union of Family
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds A_\beta\) Definition of Set Union
\(\, \ds \lor \, \) \(\ds x\) \(\in\) \(\ds B_\beta\)
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \bigcup_{\alpha \mathop \in I} A_\alpha\) Set is Subset of Union of Family
\(\, \ds \lor \, \) \(\ds x\) \(\in\) \(\ds \bigcup_{\alpha \mathop \in I} B_\alpha\) Set is Subset of Union of Family
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}\) Definition of Set Union

Thus by definition of subset:

$\ds \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha} \subseteq \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}$

$\Box$


\(\ds x\) \(\in\) \(\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}\)
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \bigcup_{\alpha \mathop \in I} A_\alpha\) Definition of Set Union
\(\, \ds \lor \, \) \(\ds x\) \(\in\) \(\ds \bigcup_{\alpha \mathop \in I} B_\alpha\)
\(\ds \leadsto \ \ \) \(\ds \exists \beta \in I: \, \) \(\ds x\) \(\in\) \(\ds A_\beta\) Definition of Union of Family
\(\ds \exists \beta \in I: \, \) \(\, \ds \lor \, \) \(\ds x\) \(\in\) \(\ds B_\beta\)
\(\ds \leadsto \ \ \) \(\ds \exists \beta \in I: \, \) \(\ds x\) \(\in\) \(\ds A_\beta \cup B_\beta\) Definition of Union of Family
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha}\)

Thus by definition of subset:

$\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha} \subseteq \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha}$

$\Box$


By definition of set equality:

$\ds \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}$

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Set_Union_is_Self-Distributive/Families_of_Sets&oldid=578493"