I am currently taking discrete math, and we have been learning several math symbols that we have used in our proof-writing assignments. Obviously, we have discussed the $\in$ symbol for inclusion in a set, but we have never mentioned the $\ni$ symbol to show that a set contains an element, i.e. $A\ni a$.

In several instances I have found that I could state something more concisely by using this symbol. For example, in a current homework that I am working on, I would like to say that "$R_1,R_2\ni(a,b)$", where $R_1$ and $R_2$ are equivalence relations. This is more concise than saying "$(a,b)\in R_1$ and $(a,b)\in R_2$".

However, I don't see this symbol very frequently and I would prefer to make sure that it is commonly acceptable to use this symbol even when there is another way to state something like this. Is the $\ni$ symbol commonly accepted in mathematics or is the $\in$ symbol generally preferred?

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    $\begingroup$ I see no issue in using it. I know that I've used it on occasion myself here, but certainly far less often than I use $\in$. If you are too concerned about it not being kosher then perhaps use words instead. E.g. "$R_1$ and $R_2$ are equivalence relations which both contain $(a,b)$." $\endgroup$ – JMoravitz Apr 15 '19 at 19:55
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    $\begingroup$ Why not $(a,b)\in R_1,R_2$? However, I have seen the notation $A\ni a$, refering to precisely the same as $a\in A$, but rarely to be honest. $\endgroup$ – mrtaurho Apr 15 '19 at 19:56
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    $\begingroup$ @mrtaurho the order in which things are written can sometimes imply which terms or phrases have the most influence and importance in a phrase and where your attention should be drawn the most. $R_1,R_2\ni (a,b)$ to me makes it seem that it is the fact that we are talking about $R_1$ and $R_2$ having this property that is what is important and are what are special as opposed to $(a,b)\in R_1,R_2$ which makes it seem that $(a,b)$ is special for satisfying this condition. It doesn't come up nearly as often in English but in languages like Japanese it plays a heavy role in nuanced meanings. $\endgroup$ – JMoravitz Apr 15 '19 at 20:07
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    $\begingroup$ In general, it is much more important to be clear than to be concise. The phrase "$R_1,R_2 \ni a$" [or even "$a \in R_1,R_2$"] may cause some readers confusion; is $a$ assumed to be in $R_1$ and $R_2$ something else sort of like $R_1$, or is $a$ in both. So you could say $R_1 \cap R_2 \ni a$ or $a \in R_1 \cap R_2$; both would be better. In general, write so that there is no doubt what you mean, even if it means a longer or an extra sentence. $\endgroup$ – Mike Apr 15 '19 at 20:29
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    $\begingroup$ In general, if you want to make it easy for your reader to get through your writing, always try to use a flow close to natural language, and try to use standard notation. The more "custom" notation you use the harder it becomes to read, but at the same time it's not saving the writer a whole lot $\endgroup$ – NazimJ Apr 15 '19 at 20:50

As is reflected in the comments, it's better to avoid nonstandard notation unless you have a very good reason to use it. Deleting an "and" from your sentence is not a good enough reason, in my opinion.

I would go a step further and say that you should avoid even standard symbols unless you have a pretty good reason to use them. (Notice the drop from "very good reason" to "pretty good reason".)

For example:

The elements $a$ and $b$ are related under both $R_1$ and $R_2$.

I think this sentence is just as precise as anything involving "$\in$", but it is much more clear because it flows naturally in English and doesn't require the reader to unpack the meaning of any symbols. Moreover, it unobtrusively reminds the reader what kind of objects $a$, $b$, $R_1$, and $R_2$ are.


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