Difference between revisions of "Ring of sets"
From Maths
(Created page with "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 alge...") |
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| + | 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. | ||
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| + | Take any <math>A,B\in R_1\cap R_2</math> | ||
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| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
Revision as of 20:01, 16 March 2015
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][1] 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
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
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]
References
- ↑ Page 19 - Halmos - Measure Theory - Springer - Graduate Texts in Mathematics (18)
