I've seen expressions like $x\lt y\in S$ used on this site to mean "$x\lt y$ and $x\in S$ and $y\in S$." This has me worried because I sometimes use $x\lt y\in S$ to abbreviate "$x\lt y$ and $y\in S$". How widespread is the former usage? Should I avoid using abbreviations such as $x\lt y\in S$ and $x\ne y\in S$ etc. because I'm likely to be misunderstood?

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    $\begingroup$ It's ambiguous. It's terrible notation. $S$ could be a set of ordered pairs and $\lt$ could be a relation thereon. You should avoid it at the cost of a few more keystrokes. $\endgroup$ – David Roberts Dec 23 '20 at 2:44
  • $\begingroup$ Since you asked how widespread it is, here is search for $x\lt y\in S$ on SearchOnMath with the domain limited to MO. (You can also change to other domains - just to have some comparison.) Already on the first page, it shows many different expressions. (I am not sure how exactly this search engine decides which two expressions are similar.) Here is the same search on ApproachZero. (A0 indexes Mathematics and AoPS, but not MO.) $\endgroup$ – Martin Sleziak Dec 23 '20 at 8:33
  • $\begingroup$ @bof Of course, in both cases - in SearchOnMath and in Approach Zero you can use this as a search query, too. (I am not sure to which extent this could be helpful. But at least among the search results you can see some occurrences - still, you'd have to check them individually to see what meaning exactly the author of the post has in mind. As you said in the last comment, for this you can't really rely on computer.) $\endgroup$ – Martin Sleziak Dec 23 '20 at 14:24
  • $\begingroup$ @bof all three of your examples are unambiguous because of how those relation work. The problem with $x\lt y \in S$ is, as you pointed out, is that people can quite reasonably assume it means one of two mutually exclusive things. $\endgroup$ – David Roberts Dec 23 '20 at 22:42
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    $\begingroup$ I’m voting to close this question because it should be posted on main and not on meta. $\endgroup$ – Federico Poloni Jan 3 at 11:18
  • $\begingroup$ @bof Your question does not apply only to writing mathematics on the main site, but to writing mathematics in general. In its current version, you didn't even write "on MO" in it (and even if you would that would be a boat-programming specification). Anyhow, we have other users and moderators to take this decision; this is just my opinion and I do not claim I know more. :) $\endgroup$ – Federico Poloni Jan 3 at 13:23

Yes, ambiguous notation should be avoided in places with a general audience, like journals and here on MathOverflow. It's useful to have shorthand in specific places where all readers can agree on the meaning but it is evidently incorrect to assume that everyone everywhere agrees on that same meaning.


The short versions of your two suggested meanings are: $$x,y\in S,\, x<y$$ or $$y\in S,\, x<y$$ Neither require much more typing or space than the original ambiguous expresion.


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