Definition:Algebra of Sets/Definition 1

Definition

Let $X$ be a set.

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

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

Then $\RR$ is an algebra of sets over $X$ if and only if the following conditions hold:

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

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