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 $R$ of sets such that:

• $$\forall A\in R\forall B\in R(A\cup B\in R)$$
• $$\forall A\in R\forall B\in R(A-B\in R)$$

A ring that exists

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"

First theorems

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The empty set belongs to every ring

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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

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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:

• $$A,B\in R_1$$
• $$A,B\in R_2$$

This means:

• $$A\cup B\in R_1$$ as $R_1$ is a ring
• $$A-B\in R_1$$ as $R_1$ is a ring
• $$A\cup B\in R_2$$ as $R_2$ is a ring
• $$A-B\in R_2$$ as $R_2$ is a ring

But then:

• As $$A\cup B\in R_1$$ and $$A\cup B\in R_2$$ we have $$A\cup B\in R_1\cap R_2$$
• As $$A- B\in R_1$$ and $$A- B\in R_2$$ we have $$A- B\in R_1\cap R_2$$

Thus $$R_1\cap R_2$$ is a ring.