A family of sets is simply a set of sets. For example: .
A union of a family of sets consists of all elements of all its elements. . In our example: .
An intersection of a family of sets consists of all elements appearing in every its element. . In our example: .
We can define an ordered pair of elements as . A product of two given sets is the set . Set is usually denoted as .
For a given set , the family consists of all subsets of . E.g. for , we get .
A subset is called a relation on . The fact that is usually denoted as . As for any other sets we can consider a union or intersection of two relations. We often also consider some properties of relations. E.g. a relation on is reflexive, if for any , . Notice that giver two reflexive , their intersection is also reflexive, because for all , and , so also .