Difference between revisions of "Ring of sets"

Latest revision as of 17:21, 18 August 2016

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

A Ring of sets is a non-empty class $R$ ^[1] of sets such that:

Take a set $X$ , the power set of $X$ , $\mathcal{P}(X)$ is a ring (further still, an algebra) - the proof of this is trivial.

This ring is important because it means we may talk of a "ring generated by"

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The empty set belongs to every ring

Take any $A\in R$ then $A-A\in R$ but $A-A=\emptyset$ so $\emptyset\in R$

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Given any two rings, $R_1$ and $R_2$ , the intersection of the rings, $R_1\cap R_2$ is a ring

We know $\emptyset\in R$ , this means we know at least $\{\emptyset\}\subseteq R_1\cap R_2$ - it is non empty.

Take any $A,B\in R_1\cap R_2$ (which may be the empty set, as shown above)

Then:

This means:

But then:

Thus $R_1\cap R_2$ is a ring.

@@ Line 2: / Line 2: @@
 Note that every [[Algebra of sets]] is also a ring, and that an [[Algebra of sets]] is sometimes called a '''Boolean algebra'''
-==Definition==
+==[[/Definition|Definition]]==
-A Ring of sets is a non-empty class {{M|R}}<ref>Page 19 - Halmos - Measure Theory - Springer - Graduate Texts in Mathematics (18)</ref> of sets such that:
+{{/Definition}}
-* <math>\forall A\in R\forall B\in R(A\cup B\in R)</math>
-* <math>\forall A\in R\forall B\in R(A-B\in R)</math>
 ==A ring that exists==
 Take a set {{M|X}}, the [[Power set|power set]] of {{M|X}}, {{M|\mathcal{P}(X)}} is a ring (further still, an [[Algebra of sets|algebra]]) - the proof of this is trivial.