Definition:Ring of Sets

From ProofWiki
Jump to navigation Jump to search

This page is about rings of sets. For other uses, see Definition:Ring.

Definition

Definition 1

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

\((RS \, 1_1)\)   $:$   Non-Empty:    \(\displaystyle \mathcal R \ne \varnothing \)             
\((RS \, 2_1)\)   $:$   Closure under Intersection:      \(\displaystyle \forall A, B \in \mathcal R:\) \(\displaystyle A \cap B \in \mathcal R \)             
\((RS \, 3_1)\)   $:$   Closure under Symmetric Difference:      \(\displaystyle \forall A, B \in \mathcal R:\) \(\displaystyle A * B \in \mathcal R \)             


Definition 2

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

\((RS \, 1_2)\)   $:$   Empty Set:    \(\displaystyle \varnothing \in \mathcal R \)             
\((RS \, 2_2)\)   $:$   Closure under Set Difference:      \(\displaystyle \forall A, B \in \mathcal R:\) \(\displaystyle A \setminus B \in \mathcal R \)             
\((RS \, 3_2)\)   $:$   Closure under Union:      \(\displaystyle \forall A, B \in \mathcal R:\) \(\displaystyle A \cup B \in \mathcal R \)             


Definition 3

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

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


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.