Axiom:Ring of Sets Axioms/Axioms 1

Definition

Let $\RR$ be a system of sets.

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

\((\text {RS} 1_1)\)	$:$		Non-Empty:		\(\ds \RR \ne \O \)
\((\text {RS} 2_1)\)	$:$		Closure under Intersection:	\(\ds \forall A, B \in \RR:\)	\(\ds A \cap B \in \RR \)
\((\text {RS} 3_1)\)	$:$		Closure under Symmetric Difference:	\(\ds \forall A, B \in \RR:\)	\(\ds A \symdif B \in \RR \)

These criteria are called the ring of sets axioms.

Also see

• Definition:Ring of Sets/Definition 1

• Equivalence of Definitions of Ring of Sets