Sets:Families

A set of sets is usually referred to as a family or collection of sets. Often, families of sets are written with either a script or Fraktur font to easily distinguish them from other sets. For a family of sets ${\mathfrak {A}}$ , define the union and intersection of the family by,

${\begin{aligned}\bigcup {\mathfrak {A}}&=\bigcup _{A\in {\mathfrak {A}}}A=\{x:x{\text{ is in some }}A\in {\mathfrak {A}}\}\\\bigcap {\mathfrak {A}}&=\bigcap _{A\in {\mathfrak {A}}}A=\{x:x{\text{ is in all }}A\in {\mathfrak {A}}\}\end{aligned}}$

For a family of sets, we say that it is pairwise disjoint if any two distinct sets we choose from the family are disjoint.

Resources

• Course Textbook, pages 109-115
• Indexed Families of Sets, Mathematics LibreTexts