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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?

mrtaurho
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csch2
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    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 $(a,b)$." – JMoravitz Apr 15 '19 at 19:55
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    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. – mrtaurho Apr 15 '19 at 19:56
  • @mrtaurho That certainly works. However in particular contexts I would still prefer to write it in the other direction to more clearly convey what I am trying to say, so I am still curious whether the $\ni$ symbol is less preferred. – csch2 Apr 15 '19 at 20:03
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    @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. – JMoravitz Apr 15 '19 at 20:07
  • @JMoravitz Fair point. I only was slightly confused why, according to the OP, $R_1,R_2\ni(a,b)$ is a valid option but we have to write "$(a,b)\in R_1$ and $(a,b)\in R_2$". I agree with both of you that the order might be of relevance in particular situations. On the other hand it seems to me aswell that $\in$ is preferred over $\ni$. – mrtaurho Apr 15 '19 at 20:28
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    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. – Mike Apr 15 '19 at 20:29
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    On that note I was taught to be very careful using relations in sentences: e.g., my old college professors told me that 'We know that $a<b$' is stylistically bad and 'We know that $a$ and $b$ satisfy the inequality $a<b$' is much better. – Mike Apr 15 '19 at 20:34
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    Why not: "The relations $R_1$ and $R_2$ both contain the ordered pair $(a,b)$." It reads more like natural English and affords you the opportunity to remind the reader what kinds of objects $R_1$, $R_2$, and $(a,b)$ are. Even better might be "The elements $a$ and $b$ are related under both $R_1$ and $R_2$". – Austin Mohr Apr 15 '19 at 20:45
  • I use the symbol only in the context $f: X \ni x \mapsto f(x) \in Y$, which allows to specify the domain, the formula and the co-domain in one expression. I've never seen it used in a different context. – Adam Latosiński Apr 15 '19 at 20:47
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    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 – NazimJ Apr 15 '19 at 20:50
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    @AustinMohr That is a good suggestion. In general it seems that the consensus is "err on the side of wordiness in order to provide clarity and avoid cumbersome notation". – csch2 Apr 15 '19 at 20:54
  • @csch2 It's good that you're already thinking about issues of style. Your writing will be greatly improved in the long run, while many of your peers will probably never develop their own voice. Good luck with your studies. – Austin Mohr Apr 15 '19 at 21:01
  • As others have mentioned clarity is a big issue. I personally have seen and do use "$\ni$" as a symbol to represent "such that," but have seen others use it as you have. Here is a comment and post that mention this, https://math.stackexchange.com/a/309516/644411, https://math.stackexchange.com/q/15455/644411, which if you're interested can provide you more information about the symbol's use . – Hotdog Apr 21 '19 at 20:15

2 Answers2

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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.

Austin Mohr
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Given what you wrote, all of the following lines, written in several different syntactical notations, would have the same semantic meaning:

SYNTAX Name of Language
R₁, R₂ ∋ a, b Rendering of Stack Exchange Mark-down Language
a, b ∈ R₁, R₂ Rendering of Stack Exchange Mark-down Language
$a,b \in R_{1},R_{2}$ Rendering of LaTeX
$R_{1},R_{2} \ni a,b$ Rendering of LaTeX
*R***₁**, *R*₂ ∋ *a*, *b* Source Code for Stack Exchange Mark-down Language
*a*, *b* ∈ *R***₁**, *R*₂ Source Code for Stack Exchange Mark-down Language
$a,b \in R_{1},R_{2}$ Source Code for LaTeX
$R_{1},R_{2} \ni a,b$ Source Code for LaTeX

However, all of the following symbols are used by North American Mathematicians for the English phrase "such that".

  • $∍$
  • ∍
  • $\backepsilon$
  • \backepsilon
  • $\ni$
  • ∋
  • \ni

Using the symbol ∋ for "element of" or "contains" is frowned upon by many mathematicians living in Canada and the United States of America because of its use as a short-hand notation for the English phrase "such that". More and more, people are using colons : to represent the English phrase "such that", but older mathematicians sometimes tell you that writing ∍ is more appropriate than writing a colon :.

Examples of mathematical definitions for a graph theoretical path which use the symbols : and ∍ are shown below along with the same mathematical definition with in English.

One Definition of a Path
A directed path is a graph $P = (VS, E)$ such that there exists vertices $V, W \in VS$, not necessarily distinct, such that the edge set $E$ represents a function $VS \setminus \{W\} \to VS \setminus \{V\}$ with the following properties:

$\forall n \in \mathbb{N}$ and $\forall X \in VS$, we have $E^{n}(X) \neq V$.

$\forall X \in VS \quad \exists k \in \mathbb{N} : E^{k}(X) = W$
A Definition of a Path in Symbolic Notation using the symbol ∍
Def directed path as $(VS, E) ∍ \exists V, W \in VS ∍ E ∍ VS \setminus \{W\} \to VS \setminus \{V\} \land \forall n \in \mathbb{N} \land \forall X \in VS, E^{(n)}(X) \neq V \land \forall X \in VS \quad \exists k \in \mathbb{N} ∍ E^{(k)}(X) = W$
A Definition of a Path in Symbolic Notation using the symbol :
Def directed path as $(VS, E) : \exists V, W \in VS : E : VS \setminus \{W\} \to VS \setminus \{V\} \land \forall n \in \mathbb{N} \land \forall X \in VS, E^{(n)}(X) \neq V \land \forall X \in VS \quad \exists k \in \mathbb{N} : E^{(k)}(X) = W$
A Definition of a Path using Mostly English Words
Define directed path as an ordered pair $VS$, $E$ such that there exists vertex $V$ and vertex $W$ in $VS$ such that $E$ is a mapping from $VS$ without $W$ to $VS$ without $V$ and for any $n$ in the natural numbers $\mathbb{N}$ and for any vertex $X$ in $VS$, the $n^{\text{th}}$ iteration of mapping $E$ applied to vertex $X$ is not equal to $V$ and for any $X$ in vertex set $VS$ there exists at least one $k$ in the set of all natural numbers such that the $k^{\text{th}}$ iteration of mapping $E$ applied to vertex $X$ is equal to vertex $W$.
Another Definition of a Path using Pure English Words
A directed path is a graph having a vertex set and an edge set such that there exists a first vertex and a final vertex such that for every node, there is a way to get to the last vertex from that node and for every node, there is no way to get to the last vertex and every vertex, except for the final vertex, has exactly one outgoing edge.
Toothpick Anemone
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