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 $\mathcal R \subseteq \powerset X$ 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.