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Can I use the following definition?

Def.. let be $A$ a set and $B \in \mathscr{P}(\mathscr{P}(A))$, $(A,B)$ is algebra of sets if

  • $\emptyset \in B$

  • $\forall X \in B( (A-X) \in B)$

  • $\forall X,Y \in B((X \cap Y ) \in B)$

It is correct? Thanks in advance!

mle
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  • You should seriously consider writing $X \in \mathcal{P}(Y)$ as $X \subseteq Y$. Writing the former is akin to writing $x = \ln e^y$. – M. Vinay Jun 06 '14 at 15:27

1 Answers1

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If by "is it correct" you mean "is it equivalent to the normal definition" (closed under pairwise intersections, pairwise unions, contains $\emptyset$ and $A$) then the answer is "yes."

The second bullet asserts it is closed under complements and the third under intersections.

Suppose $S,T\in B$. Since $S\cup T=A\setminus((A\setminus S)\cap (A\setminus T))$, and the right hand side consists of complements and intersections of things in $B$, $S\cup T$ is in $B$ as well.

Finally, $A\setminus\emptyset=A$ is in $B$ also, so you have both identities.

rschwieb
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