Family of Sets

Definition

A collection of any sets

Operations

Union

$\bigcup F=\bigcup _{A\in F}A=\{x|\exists A\in F,x\in A\}$ $\bigcup F=\bigcup _{i\in I}{A}_i=\{x|\exists i\in I,x\in A_i\}$ where $F$ is an Indexed Family with index set $I$

Intersection

$\bigcap F=\bigcap _{A\in F}A=\{x|\forall A\in F,x\in A\}$ $\bigcap F=\bigcap _{i\in I}{A}_i=\{x|\forall i\in I,x\in A_i\}$ where $F$ is an Indexed Family with index set $I$

Cartesian Product

$\prod A_i=\{({\alpha }_i{)}_{i\in I}|\forall i\in I,{\alpha }_i\in A_i\}$

Facts

$(\bigcup _{A\in F}A{)}^{\complement }=\bigcap _{A\in F}{A}^{\complement }$

$(\bigcap _{A\in F}A{)}^{\complement }=\bigcup _{A\in F}{A}^{\complement }$

$A\cap (\bigcup _{B\in F}B)=\bigcup _{B\in F}(A\cap B)$

$A\cup (\bigcap _{B\in F}B)=\bigcap _{B\in F}(A\cup B)$