## Structures of set theory

We begin by recalling the basic notions of set theory, which should be familiar to everybody:

Definition 1.1:

Let ${\displaystyle S,T\subseteq U}$ be subsets of some universal set ${\displaystyle U}$.

• The union of ${\displaystyle S}$ and ${\displaystyle T}$ is ${\displaystyle S\cup T:=\{u\in U|u\in S\vee u\in T\}}$.
• The intersection of ${\displaystyle S}$ and ${\displaystyle T}$ is ${\displaystyle S\cap T:=\{u\in U|u\in S\wedge u\in T\}}$.
• The sum of ${\displaystyle S}$ and ${\displaystyle T}$ is ${\displaystyle S+T:=(S\cup T)\setminus (S\cap T)}$.

We will follow the convention to denote the intersection of ${\displaystyle S}$ and ${\displaystyle T}$ just by juxtaposition: ${\displaystyle ST:=S\cap T}$.

Given a universal set ${\displaystyle U}$, the subsets of ${\displaystyle U}$ have the algebraic structure of a ring:

Theorem 1.2:

Let ${\displaystyle U}$ be a universal set. Let ${\displaystyle R:={\mathcal {P}}(U)}$, the power set of ${\displaystyle U}$.