View source for 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 {{M|R}}<ref>Page 19 - Halmos - Measure Theory - Springer - Graduate Texts in Mathematics (18)</ref> 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>

==A ring that exists==
Take a set {{M|X}}, the [[Power set|power set]] of {{M|X}}, {{M|\mathcal{P}(X)}} is a ring (further still, an [[Algebra of sets|algebra]]) - the proof of this is trivial. 

This ring is important because it means we may talk of a "[[Ring generated by|ring generated by]]"

==First theorems==
{{Begin Theorem}}
The empty set belongs to every ring
{{Begin Proof}}
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>
{{End Proof}}
{{End Theorem}}
{{Begin Theorem}}
Given any two rings, {{M|R_1}} and {{M|R_2}}, the intersection of the rings, {{M|R_1\cap R_2}} is a ring
{{Begin Proof}}
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> (which may be the empty set, as shown above)

Then:
* <math>A,B\in R_1</math>
* <math>A,B\in R_2</math>


This means:
* <math>A\cup B\in R_1</math> as {{M|R_1}} is a ring
* <math>A-B\in R_1</math> as {{M|R_1}} is a ring
* <math>A\cup B\in R_2</math> as {{M|R_2}} is a ring
* <math>A-B\in R_2</math> as {{M|R_2}} is a ring

But then:
* As <math>A\cup B\in R_1</math> and <math>A\cup B\in R_2</math> we have <math>A\cup B\in R_1\cap R_2</math>
* As <math>A- B\in R_1</math> and <math>A- B\in R_2</math> we have <math>A- B\in R_1\cap R_2</math>

Thus <math>R_1\cap R_2</math> is a ring.
{{End Proof}}
{{End Theorem}}



==References==

{{Definition|Measure Theory}}