Definition:Algebra of Sets/Definition 1
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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 over $S$ if and only if $\RR$ satisfies the algebra of sets axioms:
\((\text {AS} 1)\) | $:$ | 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 $S$: | \(\ds \forall A \in \RR:\) | \(\ds \relcomp S A \in \RR \) |
Also see
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $1.1$: Algebras and Sigma-Algebras