2. Families of sets and relations

A family of sets is simply a set of sets. For example: \mathcal{A}=\{\{\varnothing\},\{\varnothing,\{\varnothing\},\{\{\varnothing\}\}\},\{\varnothing,\{\{\varnothing\}\}\}\}.

A union of a family of sets consists of all elements of all its elements. \bigcup\mathcal{A}=\{a\colon \exists_{A\in\mathcal{A}}a\in A\}. In our example: \bigcup\mathcal{A}=\{\varnothing,\{\varnothing\},\{\{\varnothing\}\}\}.

An intersection of a family of sets consists of all elements appearing in every its element. \bigcap\mathcal{A}=\{a\colon \forall_{A\in\mathcal{A}}a\in A\}. In our example: \bigcap\mathcal{A}=\{\varnothing\}.

We can define an ordered pair of elements a,b as \left<a,b\right>=\{\{a\},\{a,b\}\}. A product of two given sets A,B is the set A\times B=\{\left<a,b\right>\colon a\in A,b\in B\}. Set A\times A is usually denoted as A^2.

For a given set A, the family \mathcal{P}(A) consists of all subsets of A. E.g. for A=\{1,2,3\}, we get \mathcal{P}(A)=\{\varnothing, \{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}.

A subset r\subseteq A^2 is called a relation on A. The fact that \left<a,b\right>\in r is usually denoted as a r b. 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 r on A is reflexive, if for any a\in A, a r a. Notice that giver two reflexive r, r', their intersection r\cap r' is also reflexive, because for all a\in A, a r a and a r' a, so also \left<a,a\right>\in r\cap r'.