# Definition:Algebra of Sets

## Definition

### Definition 1

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\RR \subseteq \powerset S$ be a set of subsets of $S$.

$\RR$ is an algebra of sets over $S$ if and only if $\RR$ satisfies the algebra of sets axioms:

 $(\text {AS} 1)$ $:$ Unit: $\ds S \in \RR$ $(\text {AS} 2)$ $:$ Closure under Union: $\ds \forall A, B \in \RR:$ $\ds A \cup B \in \RR$ $(\text {AS} 3)$ $:$ Closure under Complement Relative to $S$: $\ds \forall A \in \RR:$ $\ds \relcomp S A \in \RR$

### Definition 2

An algebra of sets is a ring of sets with a unit.

## Also see

• Results about Algebras of Sets can be found here.