Axiom:Algebra of Sets Axioms
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Definition
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 if and only if the following axioms hold:
\((\text {AS} 1)\) | $:$ | 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 $S$: | \(\ds \forall A \in \RR:\) | \(\ds \relcomp S A \in \RR \) |
These criteria are called the algebra of sets axioms.