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I'm reading through some notes which give a proof by induction for Bool's inequality, and I wondered if someone could explain what the notation. I'm confused by the $E^{(k)}$, what does this mean? $$ \begin{align*} \Pr\left[\bigcup_{i=1}^{k+1} E_i\right] &= \Pr\left[E^{(k)} \cup E_{k+1}\right] \\ &\leq \Pr[E^{(k)}] + \Pr[E^{(k+1)}] \\[4pt] &\leq \left(\sum_{i=1}^k \Pr[E_i]\right) + \Pr[E_{k+1}] \\ &= \sum_{i=1}^{k+1} \Pr[E_i]. \end{align*}$$

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In the context of the formula you linked to, it means $\displaystyle\bigcup_{i=1}^k E_k$.

This is not really a standard bit of notation, so I hope it is explained before it is used.

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Thank you, do the brackets around the k have any meaning? Is it that which implies the union? – superbriggs Dec 29 '12 at 16:27
It could well be at an early part of the notes, I'm only taking reading through a few pages from this source. Thank you very much – superbriggs Dec 29 '12 at 16:29
The brackets are to distinguish it from $E^k$, which has a pretty standard meaning of direct product $E\times E\times\cdots\times E$, as in $\mathbb{R}^3$. – André Nicolas Dec 29 '12 at 16:29

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