Definition:Ring of Sets/Definition 3

Definition

A system of sets $\RR$ is a ring of sets if and only if $\RR$ satisfies the ring of sets axioms:

$(\text {RS} 1_3)$	$:$	Empty Set:		$\ds \O \in \RR $
$(\text {RS} 2_3)$	$:$	Closure under Set Difference:	$\ds \forall A, B \in \RR:$	$\ds A \setminus B \in \RR $
$(\text {RS} 3_3)$	$:$	Closure under Disjoint Union:	$\ds \forall A, B \in \RR:$	$\ds A \cap B = \O \implies A \cup B \in \RR $

\((\text {RS} 1_3)\)	$:$	Empty Set:		\(\ds \O \in \RR \)
\((\text {RS} 2_3)\)	$:$	Closure under Set Difference:	\(\ds \forall A, B \in \RR:\)	\(\ds A \setminus B \in \RR \)
\((\text {RS} 3_3)\)	$:$	Closure under Disjoint Union:	\(\ds \forall A, B \in \RR:\)	\(\ds A \cap B = \O \implies A \cup B \in \RR \)