Axiom:Semiring of Sets Axioms/Axioms 2

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Definition

Let $\SS$ be a system of sets.


$\SS$ is a semiring of sets if and only if $\SS$ satisfies the following axioms:

\((1)\)   $:$   \(\ds \O \in \SS \)      
\((2)\)   $:$   $\cap$-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 $A_1, A_2, A_3, \ldots, A_n \in \SS : \ds A \setminus B = \bigcup_{k \mathop = 1}^n A_k$      

These criteria are called the semiring of sets axioms.


Also see