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This may be a really simple question to answer, but I need to know, with references to prove my affirmation, that the actual notation for "such that" on a set builder notation is the colon or vertical line, but not the oblique line. Can anybody help me with some reference for this notation? Thank you very very much.

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    $\begingroup$ You want references for that? Set builder notation is $\textbf{notation}$: i.e., if you want to use a different symbol, go for it. $\endgroup$ – Don Thousand Dec 14 '18 at 3:42
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    $\begingroup$ There are many cases where people use nonstandard notation. As long as it is established before hand that the people are using the same notation it is okay. Rather then "actual notation" it might be better to say "standard notation" or "most commonly used notation" $\endgroup$ – Q the Platypus Dec 14 '18 at 3:44
  • $\begingroup$ Thank you for your comments. I was serching for references that establishes the vertical line or colon as standard notation, and that gives some reason for not using "/" instead of the other ones. This notation is going to be use on a very elementary material for school students. $\endgroup$ – Irene Dec 14 '18 at 11:47
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    $\begingroup$ You could make up reasons why $/$ is a bad choice, but you could just as well make up such reasons for $:$ and $|$. The real reason is just that it's the notation that everyone else uses. In any case, it seems rather bizarre that you need a reference to "prove" this, especially if this is just for students. $\endgroup$ – Eric Wofsey Dec 15 '18 at 22:14
  • $\begingroup$ It never occurred to me that the ":" in the weirdly-named "set-builder" notation was the mathematical notation for "such that". I suppose, then, that "{" is the notation for "the set of all", but what does "}" stand for? $\endgroup$ – bof Dec 15 '18 at 23:56
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Both $\vert$ and $:$ are used.
$\vert$ perhaps, being the most historical.
For clarity of notation, $:$ is preferred.
For example imagine using $\vert$ in the definition of $\{ x : |x| < 5 \}$.

Though I don't know the source of the recent amateurish use of /, it is to be avoided since it is not accepted, causes confusion and lacks notational clarity. For example, $\{ x/y \ / \ 1/xy < 10 \}$.

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