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: | \(\ds \forall A, B \in \RR:\) | \(\ds A \cap B = \O \implies A \cup B \in \RR \) |