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Is there some compact notation for saying "all the elements of a set"?

Just trying to save time and be concise really when:

$\forall x\in S:P(x)$ is a bit long-winded to some potential alternative:

$P(\bar S)$

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    $\begingroup$ I feel that $\forall x \in S \, : P(x)$ is the commonly accepted notation. One could use something like $P|_S \equiv \operatorname{true}$, but this looks slightly weird. $\endgroup$ – lisyarus Jun 15 '18 at 15:25
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    $\begingroup$ If you're not doing something closely related to logic, it's more common to be even more "long-winded" and to use words, rather than the symbol $\forall$. $\endgroup$ – Mark S. Jun 16 '18 at 20:12
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Sometimes people define $P(S)$ to mean $\forall x\in S, P(x)$. If you explicitly define it this way, then you can use it this way.

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