# Definition:Algebra of Sets

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## Definition

### Definition 1

Let $X$ be a set.

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

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

Then $\RR$ is an **algebra of sets over $X$** if and only if the following conditions hold:

\((\text {AS} 1)\) | $:$ | Unit: | \(\displaystyle X \in \RR \) | |||||

\((\text {AS} 2)\) | $:$ | Closure under Union: | \(\displaystyle \forall A, B \in \RR:\) | \(\displaystyle A \cup B \in \RR \) | ||||

\((\text {AS} 3)\) | $:$ | Closure under Complement Relative to $X$: | \(\displaystyle \forall A \in \RR:\) | \(\displaystyle \relcomp X A \in \RR \) |

### 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.