# Definition:Ring of Sets

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.