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I would like to express the statement:

At (1-α)100% confidence, $e \le z_{\frac{α}{2}}\sqrt{\frac{{p̂(1-p̂)}}{n}}$

But I want to express the first part using logical operators. What is the logical operator equivalent of the first part of the above expression?

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There is no such symbol similar to $\forall, \exists$ for "For all", "There exists", respectively. You could translate it into $\operatorname{Pr}\left[e \le z_{\frac \alpha2}\sqrt{\frac{\beta(1-\beta)}n}\right] \ge 1-\alpha$, where $\operatorname{Pr}$ is the Probability operator (also denoted $\mathrm P$ or $\Bbb P$). I can't directly think of anything shorter than what you wrote down that expresses what you want.

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