Set family/Intersection and union/Definition

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

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

is called the intersection of the sets, and

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

is called the union of the sets.