# Definition:Ring of Sets

## 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.