I am trying to prove additional properties of semi algebra assuming defination of semi algebra .
Definition of semi algebra:
A class $\mathcal S\subseteq\wp(X)$ is called semi algebra over $X$ if:
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- Empty set and whole set are in $\mathcal S$.
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- It is closed under finite intersection, i.e. if $A, B$ are in $\mathcal S$ then $A\cap B$ is also in $\mathcal S$.
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- Set difference of any two sets in $\mathcal S$ is finite disjoint union of elements in $\mathcal S$.
I am trying to prove (or disprove) the following properties. Can someone please help.
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- if $A$ belongs to $\mathcal S$, then so does the complement of $A$.
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- If $A$ and $B$ belong to $\mathcal S$, then so does $A\cup B$.
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- If $A$ and $B$ belong to $\mathcal S$, and $A$ is a subset of $B$, then so does $B\setminus A$.
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- If $A_1, A_2,\dots$ upto infinity belong to $\mathcal S$, then so does $ \bigcup_{i=1}^{\infty}A_i$.
Please give some hint.
Best Answer
Let it be that the sets $A,B,C$ form a partition of $X$ and let $\mathcal S:=\{\varnothing,A,B,C,X\}$.
It is not difficult to prove that $\mathcal S$ is a semi algebra.
Now try to find answers on 1),2) and 4) with $\mathcal S$ in mind.
Let $X=[0,10)$ and $\mathcal S':=\{[a,b)\mid 0\leq a\leq b\leq10\}$.
Then it can be shown that $\mathcal S'$ is a semi algebra.
Observe that $A,B\in\mathcal S'$ with $A\subseteq B$ if $A=[4,5)$ and $B=[3,6)$.
Now with this in mind try to find an answer on 3).