Ring of sets

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] 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[A-B\in R][/math]

A ring that exists

Take a set [ilmath]X[/ilmath], the power set of [ilmath]X[/ilmath], [ilmath]\mathcal{P}(X)[/ilmath] 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"

First theorems

The empty set belongs to every ring

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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

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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] (which may be the empty set, as shown above)

Then:

• [math]A,B\in R_1[/math]
• [math]A,B\in R_2[/math]

This means:

• [math]A\cup B\in R_1[/math] as [ilmath]R_1[/ilmath] is a ring
• [math]A-B\in R_1[/math] as [ilmath]R_1[/ilmath] is a ring
• [math]A\cup B\in R_2[/math] as [ilmath]R_2[/ilmath] is a ring
• [math]A-B\in R_2[/math] as [ilmath]R_2[/ilmath] is a ring

But then:

• As [math]A\cup B\in R_1[/math] and [math]A\cup B\in R_2[/math] we have [math]A\cup B\in R_1\cap R_2[/math]
• As [math]A- B\in R_1[/math] and [math]A- B\in R_2[/math] we have [math]A- B\in R_1\cap R_2[/math]

Thus [math]R_1\cap R_2[/math] is a ring.