Axiom:Ring of Sets Axioms

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Definition

Let $\RR$ be a system of sets.

Axioms 1

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

\((\text {RS} 1_1)\)   $:$   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:      \(\ds \forall A, B \in \RR:\) \(\ds A \symdif B \in \RR \)      


Axioms 2

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

\((\text {RS} 1_1)\)   $:$   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:      \(\ds \forall A, B \in \RR:\) \(\ds A \symdif B \in \RR \)      


Axioms 3

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

\((\text {RS} 1_3)\)   $:$   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:      \(\ds \forall A, B \in \RR:\) \(\ds A \cap B = \O \implies A \cup B \in \RR \)      


Also see