Algebra of sets
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An Algebra of sets is sometimes called a Boolean algebra
We will show later that every Algebra of sets is an Algebra of sets
Definition
An class [ilmath]R[/ilmath] of sets is an Algebra of sets if[1]:
- [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
References
- ↑ p21 - Halmos - Measure Theory - Graduate Texts In Mathematics - Springer - #18
