# Definition:Algebra of Sets

Jump to navigation
Jump to search

## Definition

### Definition 1

Let $X$ be a set.

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

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

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

\((AS \, 1)\) | $:$ | Unit: | \(\displaystyle X \in \mathcal R \) | |||||

\((AS \, 2)\) | $:$ | Closure under Union: | \(\displaystyle \forall A, B \in \mathcal R:\) | \(\displaystyle A \cup B \in \mathcal R \) | ||||

\((AS \, 3)\) | $:$ | Closure under Complement Relative to $X$: | \(\displaystyle \forall A \in \mathcal R:\) | \(\displaystyle \relcomp X A \in \mathcal R \) |

### 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.