Definition:Ring of Sets/Definition 1

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Definition

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:      \(\displaystyle \forall A, B \in \mathcal R:\) \(\displaystyle A * B \in \mathcal R \)             


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