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I am reading "A First Course In Probability" by Sheldon Ross, and I am having difficulty solving one of the problems in the book:

For any sequence of events $E_1$, $E_2$, $\ldots\,$, define a new sequence $F_1$, $F_2$, $\ldots\,$ such that all the $F_i$ are disjoint and the union from $i=1$ to $n$ of $E_i$ is equal to the union from $i=1$ to $n$ of $F_i$ for any natural number $n$.

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Try it for $n = 2$. The event $E_1 \cup E_2$ can be partitioned into the events $E_1$ and $E_1^c \cap E_2$. Draw a Venn diagram to understand what is going on. Thus, taking $F_1 = E_1$ and $F_2 = E_1^c \cap E_2$, you have solved the problem for $n=2$. Can you generalize from here? Further hint: take for $F_3$ everything in $E_3$ that is not already included in $E_1 \cup E_2 = F_1 \cup F_2$. – Dilip Sarwate Apr 12 '12 at 16:30
up vote 2 down vote accepted

Let $$F_i = E_i \setminus \left( \bigcup_{j = 1}^{i-1} E_j \right), $$ i.e. $F_1 = E_1$, $F_2 = E_2 \setminus E_1, F_3 = E_3 \setminus (E_1 \cup E_2), \ldots$

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