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What does $\forall x \exists y(x + y = 0)$ mean?

Does it mean "For all x there exists a y for which x + y equals zero"?

Thanks.

Ross Millikan
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user62287
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    tip: You may accept an answer that you find helpful (you can accept one answer per question) by clicking on the $\checkmark$ to the left of the answer you'd like to accept :-) – amWhy Feb 14 '13 at 21:40

2 Answers2

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Yes, it does.

In plainer, more succinct words, it means that "every number has an additive inverse".

Joe Z.
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    Also, be absolutely sure not to confuse this with $\exists y \forall x (x + y = 0)$. The two statements are subtly different, in that in the first one, the $y$ can be different for each $x$, but in the second one, the same $y$ needs to work for all $x$. – Joe Z. Feb 14 '13 at 18:26
  • Think of it as the "such that" always immediately following the variable that the $\exists$ refers to. "For all $x$ there exists a $y$ such that $x + y = 0$" is not the same as "There exists a $y$ such that for all $x$, $x + y = 0$". And "There exists a $y$ for all $x$ such that $x + y = 0$" would not be a valid transcription of a first-order logic sentence (even though syntactically it should be equivalent to $\forall x \exists y (x + y = 0)$). – Joe Z. Feb 14 '13 at 18:43
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Yup, you're correct.

However, it seems a bit clearer if it is written (as suggested in the comments) as

$$\forall x:\exists y : (x + y = 0)$$

Joe
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    I find this notation unwieldy as well, though I have seen it in Rosen's "Discrete Mathematics and Applications" book. – JavaMan Feb 14 '13 at 18:27
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    In statements in first-order logic, there is no "such that". It's implicit in the $\exists$. – Joe Z. Feb 14 '13 at 18:28
  • Such that is denoted by $\mid$, isn't it? – mrk Feb 14 '13 at 18:28
  • @Joe good to know, thanks. – Joe Feb 14 '13 at 18:29
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    I think you just pinged yourself. :P – Joe Z. Feb 14 '13 at 18:30
  • @saadtaame: No. Set builder notation ${\cdots\mid\cdots}$ is one symbol whose components cannot be picked out and used separately from the context they are defined to mean something in. (Set brackets are separately defined to mean something else when there is no $\mid$). – hmakholm left over Monica Feb 14 '13 at 18:55
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    If at all I'd prefer the colon as a separator,$\forall x\colon\exists y\colon x+y=0$ , but only because its general spacing introduces legibility – Hagen von Eitzen Feb 14 '13 at 18:56
  • We used a lower dot in CS 245 ourselves. Like this: $\forall x . \exists y . x + y = 0$ – Joe Z. Feb 14 '13 at 19:50
  • @JoeZeng: Using a dot there seems to be peculiar to (or at least a lot more common in) computer science. This dot is a common syntactic motif for variable binders in formal calculi used in CS, probably influenced above all by the $\lambda$-calculus. – hmakholm left over Monica Feb 15 '13 at 19:24
  • @HenningMakholm my undergraduate discrete math book uses $|$ for "such that". In general I think $|$ can be a valid alternative to $:$ for "such that". (See this article which lists both for "such that") – David Etler Apr 12 '15 at 09:28