Definition:Algebra of Sets

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Definition 1

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\RR \subseteq \powerset S$ be a set of subsets of $S$.

$\RR$ is an algebra of sets over $S$ if and only if $\RR$ satisfies the algebra of sets axioms:

\((\text {AS} 1)\)   $:$   Unit:    \(\ds S \in \RR \)      
\((\text {AS} 2)\)   $:$   Closure under Union:      \(\ds \forall A, B \in \RR:\) \(\ds A \cup B \in \RR \)      
\((\text {AS} 3)\)   $:$   Closure under Complement Relative to $S$:      \(\ds \forall A \in \RR:\) \(\ds \relcomp S A \in \RR \)      

Definition 2

An algebra of sets is a ring of sets with a unit.

Also see

  • Results about algebras of sets can be found here.