Axiom:Semiring of Sets Axioms
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Definition
Let $\SS$ be a system of sets.
Axioms 1
$\SS$ satisfies the semi-ring of sets axioms if and only if:
\((1)\) | $:$ | \(\ds \O \in \SS \) | |||||||
\((2)\) | $:$ | $\cap$-stable | \(\ds \forall A, B \in \SS:\) | \(\ds A \cap B \in \SS \) | |||||
\((3)\) | $:$ | \(\ds \forall A, A_1 \in \SS : A_1 \subseteq A:\) | $\exists n \in \N$ and pairwise disjoint sets $A_2, A_3, \ldots, A_n \in \SS : \ds A = \bigcup_{k \mathop = 1}^n A_k$ |
Axioms 2
$\SS$ satisfies the semi-ring of sets axioms if and only if:
\((1)\) | $:$ | \(\ds \O \in \SS \) | |||||||
\((2)\) | $:$ | $\cap$-stable | \(\ds \forall A, B \in \SS:\) | \(\ds A \cap B \in \SS \) | |||||
\((3')\) | $:$ | \(\ds \forall A, B \in \SS:\) | $\exists n \in \N$ and pairwise disjoint sets $A_1, A_2, A_3, \ldots, A_n \in \SS : \ds A \setminus B = \bigcup_{k \mathop = 1}^n A_k$ |