Definition:Ring of Sets
Jump to navigation
Jump to search
This page is about Ring of Sets in the context of Set Theory. For other uses, see Ring.
Definition
Definition 1
A system of sets $\RR$ is a ring of sets if and only if $\RR$ satisfies the ring of sets 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: | \(\ds \forall A, B \in \RR:\) | \(\ds A \symdif B \in \RR \) |
Definition 2
A system of sets $\RR$ is a ring of sets if and only if $\RR$ satisfies the ring of sets axioms:
\((\text {RS} 1_2)\) | $:$ | Empty Set: | \(\ds \O \in \RR \) | ||||||
\((\text {RS} 2_2)\) | $:$ | Closure under Set Difference: | \(\ds \forall A, B \in \RR:\) | \(\ds A \setminus B \in \RR \) | |||||
\((\text {RS} 3_2)\) | $:$ | Closure under Union: | \(\ds \forall A, B \in \RR:\) | \(\ds A \cup B \in \RR \) |
Definition 3
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: | \(\ds \forall A, B \in \RR:\) | \(\ds A \cap B = \O \implies A \cup B \in \RR \) |
Also defined as
Some sources neglect to suggest that a ring of sets needs to be non-empty.
Also see
- Results about rings of sets can be found here.