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'For all x' = $\forall x$, but what's a convenient shorthand for 'for some x', or 'atleast one x'? $\exists$ doesn't always seem to fit the context. For example "Find $z$ such that $xz < y$ for some $y$".

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∃ does always work: for your example, a rephrasing would be "Find $z$ such that ∃$y:xz<y$".

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  • $\begingroup$ Slightly beaten to it; not sure what the etiquette is here. $\endgroup$ Oct 1, 2014 at 5:48
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    $\begingroup$ Nobody will take any offence whatever you do. $\endgroup$
    – user180040
    Oct 1, 2014 at 5:58
  • $\begingroup$ Ok, thanks. I'm leaving it, since it mentions that ∃ always works. $\endgroup$ Oct 1, 2014 at 6:05
  • $\begingroup$ Right. And if you want to say "Find $z$ such that $xz\lt y$ for all $y$" you can write "Find $z$ such that $\forall y:xz\lt y$". But please don't write stuff like "Find $z$ such that $xz\lt y\ \forall y$" or "A man's a man $\forall$ that" or "once and $\forall$". That will just annoy people. $\endgroup$
    – bof
    Aug 4, 2015 at 22:16
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$\exists$ is fine here: "Find $z$ such that $\exists y \, xz < y$."

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There's always "∃!x" for there exists exactly one x" But I don't think there's a official symbol for "for some" at least not one that i have seen: https://en.wikipedia.org/wiki/List_of_logic_symbols

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    $\begingroup$ I think $\exists$ is an official symbol for "for some", e.g. $\exists x[x^2=2]$ means "for some $x$, $x^2=2$". $\endgroup$
    – bof
    Aug 4, 2015 at 22:20

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