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For my discrete math class, my instructor has told us that the following notation is incorrect:

$\exists x \in \mathbb{N} | x > 0 \bullet x < 3$ .

And we should instead write:

$\exists x : \mathbb{N} | x > 0 \bullet x < 3$ .

What is the difference? I assume that it is a subtle difference.

Schala
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  • Not sure what the thick dot is supposed to mean, but the colon is definitely not correct for this. Also, a $\mid$ in this context is unusual (it is more common when using set builder notation). – Tobias Kildetoft Oct 08 '13 at 19:53
  • @TobiasKildetoft Depends on the school of thought. $\exists x \in \mathbb{N}$ is strictly speaking an abbreviation in set theory, but $\exists x$ on its own is malformed in type theory. – Zhen Lin Oct 08 '13 at 19:57
  • Both are used notations, the membership notation is more common, though (see http://en.wikipedia.org/wiki/Quantification#Mathematics). They are both supposed to have the same meaning, but in a class its a good idea to follow some fixed notation. – martini Oct 08 '13 at 19:57
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    @martini wow, what an awful notation. I have never seen an actual mathematician use that notation. I suppose it might be something mainly used in certain parts of the world. – Tobias Kildetoft Oct 08 '13 at 19:59
  • I would write $\exists x \in \mathbb{N}$ such that $0<x<3$. – Tyler Clark Oct 08 '13 at 20:01
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    @TobiasKildetoft: using "$:$" rather than "$\in$" in declarations is extremely common amongst logicians and computer scientists who like to distinguish between ascribing a type to a variable and asserting that an expression belongs to a set. Superscription ($x^{\mathbb{N}}$) is also commonly used for type ascription. I am sorry if you find this "definitely not correct" or "awful", but it is a matter of scientific custom rather than ethnicity. – Rob Arthan Oct 08 '13 at 20:45

1 Answers1

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It is a syntactic convention. Your instructor appears to be insisting on the syntactic conventions of the Z notation (http://en.wikipedia.org/wiki/Z_notation). In existential and universal quantifications in Z, you read "$x : \mathbb{N} \mathrel{|} 0 < x$" as a declaration of $x$ as a member of $\mathbb{N}$ satisfying the constraint $0 < x$ and you read "$\bullet\; x < 3$" as the assertion you are making about that $x$. This is convenient when you negate formulas, because the "$\bullet$" changes meaning to suit the context (it is a conjunction in an existential and an implication in a universal).

Rob Arthan
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