Category:Algebras of Sets

From ProofWiki
Jump to navigation Jump to search

This category contains results about Algebras of Sets.
Definitions specific to this category can be found in Definitions/Algebras of Sets.

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


This category has only the following subcategory.