Difference between revisions of "Ring of sets"
From Maths
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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 R[1] of sets such that:
- ∀A∈R∀B∈R(A∪B∈R)
- ∀A∈R∀B∈R(E−F∈R)
First theorems
[Expand]
The empty set belongs to every ring
[Expand]
Given any two rings, R1 and R2, the intersection of the rings, R1∩R2 is a ring
References
- Jump up ↑ Page 19 - Halmos - Measure Theory - Springer - Graduate Texts in Mathematics (18)
