- #1

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## Homework Statement

"If ##A_1,...,A_m\in O## and ##k\in ℕ##, show that the set of points in ##Ω## (the sample space) which belong to exactly ##k## of the ##A_i## belongs to ##O## (the previous exercise is the case when ##m=2## and ##k=1##)."

## Homework Equations

__Event space:__O

##O\neq ∅##

##\text{If} \space A\in O\space \text{then} \space Ω\cap A^c \in O##

##\text{If} \space A_1,...\in O \space \text{then} \space \bigcup_{i=1}^\infty A_i \in O##

## The Attempt at a Solution

This is how I have done the case for when ##m=2## and ##k=1##:

Since ##A,B\in O##, then it follows that since ##O## is closed under the operations of finite unions, that ##A\cup B\in O##. Moreover, since ##O## is closed under the operations of finite intersecctions, then it is true that ##A\cap B \in O##. Moreover, since ##A\cap B \in O##, then ##Ω\cap (A\cap B)^c \in O##. In turn, the symmetric difference of ##A## and ##B##, ##(A\cup B)\cap (A\cap B)^c =AΔB\in O##.

This is my attempt to solve for ##m,n\in ℕ##:

Since ##A_i\in O, i\in [1,m]\ \cap ℕ##, then it follows that ##\bigcup_{i=1}^m A_i \in O##. Moreover, ##\bigcup_{i\neq j} A_i\cap A_j\in O##. Then ##(\bigcup_{i=1}^m A_i)\cap (\bigcup_{i\neq j} A_i\cap A_j)^c\in O##. I've tried defining a set ##B_j=\{A_i:\forall x,y\in Ω, \text{If} \space x\in A_i, y\in A_n, \text{then} \space x,y\notin A_i\cap A_n, n\neq i\}##, with ##1\leq j \leq k## and writing: ##(\bigcup_{j=1}^k B_j)\cap (\bigcup_{i\neq j} A_i\cap A_j)^c\in O##, but I'm not really sure that I understand the problem.