Given the collection of $n$ sets $A_{1},...,A_{n}$. How can I find the most basic collection of disjoint subsets $B_{j}$ with $\cup A_{i}=\cup B_{j}$ and consisting only of intersections and set differences of $A_{i}$, and what is the name of this collection? Basically whenever sets intersect we make a "cut". Let me be more clear with the help of the following example, let: $$A_{1}:=\{1,2,3,4\}, A_{2}:=\{2,3,4\}, A_{3}:=\{4,5\}.$$ The resulting sets should be: $$B_{1}=\{1\}, B_{2}=\{2,3\}, B_{3}=\{4\}, B_{4}=\{5\}.$$ What I have until now is $B_{i}=A_{i}\setminus \cup_{i\neq k}A_{k}, i=1,...,n$ but the intersections are missing and these seem to be more complicated to write down. For example for 3 sets $A_{1},A_{2},A_{3}$ I would need the four sets $A_{1}\cap A_{2}\cap A_{3}$, $(A_{1}\cap A_{2})\setminus A_{3}$, $(A_{1}\cap A_{3})\setminus A_{2}$, $(A_{2}\cap A_{3})\setminus A_{1}$. So do the resulting sets $B_{j}$ have a name and what is the simplest way to define them?
Given n sets, what is the name of the very trivial collection of disjoint sets consiting of intersections and set differences?
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elementary-set-theory
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1They would be called the atoms of the Boolean algebra generated by the $A_i$ – 2017-01-26
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0I think I have a solution in the case of sets $A_{i}\subset\{1,...,p\}$. Define $B_{j}:=\cap_{i:j\in A_{i}}A_{i}\setminus \cup_{i:j\notin A_{i}}A_{i}$ for $j\in\{1,...,p\}$. Now omit duplicates. – 2017-01-26