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Given a set of sets $\{A_1,A_2,\dots,A_m\}$ in the power set of some set $S$. How do we describe the set of elements in $S$ that belongs to exactly $k$ of the $A_i$'s.

I feel like i need to use the set notation to show that the above described set belongs to the event space $\mathscr{F}$ of some sample space $S$ with the assumption that $A_1,\dots,A_m$ are sets in the event space $\mathscr{F}$.

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There isn't a symbol for it, but you can write it down using intersection, complement, and unions $$ \bigcup_{I\in[m]^{(k)}} \left(\bigcap_{i\in I}A_i\cap\bigcap_{j\notin I}A_j^c\right) $$ This is a finite ($\binom{m}{k}$) union of finite ($m$) intersections of events $A_i$ or complement, so it is an event.

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  • $\begingroup$ I don't understand what is $[n]^{(k)}$? Can you elaborate a bit more on how you got the formula? $\endgroup$ – user 42493 Jun 19 at 13:15
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    $\begingroup$ $[m]^{(k)}$ is the set of all size-$k$ subset of $\{1,2,\dots,m\}$, and so any such $I$ represents the $k$ values of $i$ for which we have $A_i$, and the other $m-k$ values $j$ omitted are those we have $A_j^c$. Hence the formula $\endgroup$ – user10354138 Jun 19 at 13:28
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I think you can simply say

Consider the event $E$ of those outcomes that belong to exactly $k$ of the events $\{A_1, \ldots, A_n\}$.

Finding an elaborate way to write $E$ using boolean operations will waste your time and reading it will waste your readers'.

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    $\begingroup$ I think the objective of this elementary probability space exercise is to show this set $E$ is an event, which you have swept that completely under the carpet. $\endgroup$ – user10354138 Jun 19 at 13:06
  • $\begingroup$ @user10354138 If that is in fact the question then my answer isn't an answer. But if the OP wants to know how to talk about this set in an actual application the boolean algebra should be swept under the carpet. $\endgroup$ – Ethan Bolker Jun 19 at 13:11

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