Definition:Algebra of Sets/Definition 1

Definition

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 $

\((\text {AS} 1)\)	$:$	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 $S$:	\(\ds \forall A \in \RR:\)	\(\ds \relcomp S A \in \RR \)