Set family/Intersection and union/Definition

intersection and union of a family of sets

Let ${}M_{i}$, ${}i\in I$, be a family of subsets of a set ${}G$. Then

${}\bigcap _{i\in I}M_{i}={\left\{x\in G\mid x\in M_{i}{\text{ for all }}i\in I\right\}}\,$

is called the intersection of the sets, and

${}\bigcup _{i\in I}M_{i}={\left\{x\in G\mid {\text{ there exists some }}i\in I{\text{ with }}x\in M_{i}\right\}}\,$

is called the union of the sets.