Algebra of sets

An Algebra of sets is sometimes called a Boolean algebra

We will show later that every Algebra of sets is an Algebra of sets

Definition

An class $R$ of sets is an Algebra of sets if:

• $$[A\in R\wedge B\in R]\implies A\cup B\in R$$
• $$A\in R\implies A^c\in R$$

So an Algebra of sets is just a Ring of sets containing the entire set it is a set of subsets of!

Every Algebra is also a Ring

Since for $$A\in R$$ and $$B\in R$$ we have:

$$A-B=A\cap B' = (A'\cup B)'$$ we see that being closed under Complement and Union means it is closed under Set subtraction

Thus it is a Ring of sets

See also

• $\sigma$-algebra
• Ring of sets
• Types of set algebras