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Calculus and Beyond Homework Help
If O is an event space, show for a finite number of events--
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[QUOTE="Eclair_de_XII, post: 6045403, member: 538457"] [h2]Homework Statement [/h2] "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##)." [h2]Homework Equations[/h2] [U]Event space:[/U] 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## [h2]The Attempt at a Solution[/h2] 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. [/QUOTE]
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Calculus and Beyond Homework Help
If O is an event space, show for a finite number of events--
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