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If you are working with discrete probability distributions, then for any subset $S$ of the range, you have $$ \Pr[\mathcal{M}(x)\in S] = \sum_{s\in S} \Pr[\mathcal{M}(x)=s] \leq \sum_{s\in S} e^\varepsilon \Pr[\mathcal{M}(x')=s] = e^\varepsilon \Pr[\mathcal{M}(x')\in S] $$ and so the second definition implies the first (and thus they are equivalent). You ...


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