Ring of sets

A Ring of sets is also known as a Boolean ring

Note that every Algebra of sets is also a ring, and that an Algebra of sets is sometimes called a Boolean algebra

Definition

A Ring of sets is a non-empty class [ilmath]R[/ilmath] of sets such that:

• [math]\forall A\in R\forall B\in R(A\cup B\in R)[/math]
• [math]\forall A\in R\forall B\in R(E-F\in R)[/math]

First theorems

The empty set belongs to every ring

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Take any [math]A\in R[/math] then [math]A-A\in R[/math] but [math]A-A=\emptyset[/math] so [math]\emptyset\in R[/math]

Given any two rings, [ilmath]R_1[/ilmath] and [ilmath]R_2[/ilmath], the intersection of the rings, [ilmath]R_1\cap R_2[/ilmath] is a ring

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We know [math]\emptyset\in R[/math], this means we know at least [math]\{\emptyset\}\subseteq R_1\cap R_2[/math] - it is non empty.

Take any [math]A,B\in R_1\cap R_2[/math]