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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{A}} , define the union and intersection of the family by,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\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{align}}

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