View source for Algebra of sets

An Algebra of sets is sometimes called a '''Boolean algebra'''

We will show later that every Algebra of sets is an [[Algebra of sets]] {{Todo|what could this mean?}}

==Definition==
An class {{M|R}} of sets is an ''Algebra of sets'' if<ref>p21 - Halmos - Measure Theory - Graduate Texts In Mathematics - Springer - #18</ref>:
* <math>[A\in R\wedge B\in R]\implies A\cup B\in R</math>
* <math>A\in R\implies A^c\in R</math>

So an Algebra of sets is just a [[Ring of sets]] containing the entire set it is a set of subsets of!

==Every Algebra is also a Ring==
Since for <math>A\in R</math> and <math>B\in R</math> we have:

<math>A-B=A\cap B' = (A'\cup B)'</math> we see that being closed under [[Complement]] and [[Union]] means it is closed under [[Set subtraction]]

Thus it is a [[Ring of sets]]

==See also==
* [[Sigma-algebra|{{sigma|algebra}}]]
* [[Ring of sets]]
* [[Types of set algebras]]
==References==
<references/>
{{Definition|Measure Theory}}