Definition:Ring of Sets

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This page is about Ring of Sets in the context of Set Theory. For other uses, see Ring.

Definition

Definition 1

A ring of sets $\RR$ is a system of sets with the following properties:

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


Definition 2

A ring of sets $\RR$ is a system of sets with the following properties:

\((\text {RS} 1_2)\)   $:$   Empty Set:    \(\displaystyle \O \in \RR \)             
\((\text {RS} 2_2)\)   $:$   Closure under Set Difference:      \(\displaystyle \forall A, B \in \RR:\) \(\displaystyle A \setminus B \in \RR \)             
\((\text {RS} 3_2)\)   $:$   Closure under Union:      \(\displaystyle \forall A, B \in \RR:\) \(\displaystyle A \cup B \in \RR \)             


Definition 3

A ring of sets $\RR$ is a system of sets with the following properties:

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


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.