Difference between revisions of "Algebra of sets"
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An Algebra of sets is sometimes called a '''Boolean algebra''' | An Algebra of sets is sometimes called a '''Boolean algebra''' | ||
| − | We will show later that every Algebra of sets is an [[Algebra of sets]] | + | We will show later that every Algebra of sets is an [[Algebra of sets]] {{Todo|what could this mean?}} |
==Definition== | ==Definition== | ||
Revision as of 21:38, 19 March 2016
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 R of sets is an Algebra of sets if[1]:
- [A∈R∧B∈R]⟹A∪B∈R
- A∈R⟹Ac∈R
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 A∈R and B∈R we have:
A−B=A∩B′=(A′∪B)′ 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
References
- Jump up ↑ p21 - Halmos - Measure Theory - Graduate Texts In Mathematics - Springer - #18
