Algebra of sets

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Algebra of sets
$\mathcal{A}\subseteq\mathcal{P}(X)$ For an algebra of sets, $\mathcal{A}$ on $X$
Defining properties:
1)	$\forall A\in\mathcal{A}[A^C\in\mathcal{A}]$
2)	$\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A} ]$

Note: Every algebra of sets is a ring of sets (see below)

Definition

An algebra of sets is a collection of sets, $\mathcal{A}$ such that:

• $\forall A\in\mathcal{A}[A^C\in\mathcal{A}]$
  • In words: For all $A$ in $\mathcal{A}$ the complement of $A$ (with respect to $X$) is also in $\mathcal{A}$
• $\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}]$
  • In words: For all $A$ and $B$ in $\mathcal{A}$ their union is also in $\mathcal{A}$

Claim 1: Every algebra of sets is also a ring of sets

Immediate properties

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TODO: Do this as a list of inline theorem boxes

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• $\mathcal{A}$ is $\setminus$-closed
• $\emptyset\in\mathcal{A}$
• $X\in\mathcal{A}$
• $\mathcal{A}$ is $\cap$-closed

Proof of claims

[Expand]

Claim 1: Every algebra of sets is also a ring of sets

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This is trivial. In order to show $\mathcal{A}$ is a ring of sets we require two properties:

1. $\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}]$ - this is clearly satisfied by definition of an algebra of sets
2. $\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]$ - that is $\mathcal{A}$ must be $\setminus$-closed
   • But we've already shown this in the immediate properties section above!

This completes the proof

See also

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

OLD PAGE

An Algebra of sets is sometimes called a Boolean algebra

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

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TODO: what could this mean?

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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