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If I am talking about sets $G$ and $H$ and I want to say in words that $G\subset H$, I, like everyone else, will say that $G$ is contained in $H$, or that $H$ contains $G$.

But if I am talking about a set $G$ and a single point $x$, I get vaguely uneasy if I say that $x$ "is contained in" $G$ or that $G$ "contains" $x$. The uneasiness is connected to the idea that it would only be correct to say that $\{x\}$ is contained in $G$, and that it is an abuse of terminology to say the same of $x$ itself.

An alternative is to say that $x$ "is an element of" $G$, which I think is quite standard. But this only avails if I want to mention $x$ first. Sometimes the prose works better to put $G$ first, and this is where my problem arises.

Since "$G$ contains $x$" makes me uneasy, and "$G$ contains $\{x\}$" seems circumlocutory, I will often say that "$G$ includes $x$".

Sometimes I will even do this when $x$ comes first, and say that "$x$ is included in $G$" as a synonym for "$x$ is an element of $G$".

Is this some crazy thing that I made up myself, or is it common usage that I have unconsciously absorbed from the literature? Does everyone else say "$G$ contains $x$" in this case, or do others feel a similar unease about it?

[I should clarify that I'm not only interested in how to say this orally, but also in writing.]

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I would say '$x$ belongs to $G$'. –  Sigur Aug 18 '12 at 17:45
If you want to mention $G$ before mentioning $x$, you could say, although this is clumsy aswell, '$G$ has $x$ as an element'. Lastly, you can also say '$x$ is a point of/in $G$'. –  Olivier Bégassat Aug 18 '12 at 17:56
Strange, I feel uneasy about saying "H contains G" for $G\subset H$, I would always say "G is a subset of H". On the other hand, I would have no problem saying "G contains x". Overall, though, I think this is the sort of overloading that is common in math and the meaning has to be given by the context. –  process91 Aug 18 '12 at 18:45
"Contains" is not a technical term with a precise definition, is it? In my mind it can refer to both "contains as an element" and "contains as a subset". I don't see why you have chosen the latter as the only possible meaning. –  Rahul Aug 18 '12 at 18:50
Regarding the first sentence: not like everyone else. I consider contains too ambiguous for use. If I want to verbalize $G\subseteq H$, I say *G is a subset of H$. –  Brian M. Scott Aug 18 '12 at 19:13

3 Answers 3

up vote 13 down vote accepted

Paul Halmos (How to write mathematics, http://www.math.uga.edu/~azoff/courses/halmos.pdf) suggests to distinguish between "$G$ contains $x$" and "$G$ includes $x$", the former meaning $x\in G$ and the latter $x\subseteq G$. Mark seems to have the opposite intuition about "contains" and "includes".
This shows that Halmos's idea apparently has not caught on.

On the other hand, it is rare that it is not absolutely clear from the context whether "$G$ contains $x$" means $x\subseteq G$ or $x\in G$. And mathematics is usually communicated in writing or spoken language together with something written on the blackboard or on paper. So I think it is ok to say "$G$ contains $x$" for $x\in G$.

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+1, nice reference! I hope it will not be considered rude for me to suggest that you switch the order of "the latter meaning..." and "the former meaning...", which I found confusing at first glance. –  Rahul Aug 18 '12 at 20:22
@Rahul: I am not offended and I changed the order. –  Stefan Geschke Aug 18 '12 at 21:41

In lecturing I'd verbalize it as G "contains-element" $x$.

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In the very unlikely event that I both got trapped in a sentence with $G$ mentioned first and chose not to stop and start over, I’d say G has/contains x as an element. –  Brian M. Scott Aug 18 '12 at 19:15
@BrianM.Scott Would "G possesses x as an element" be acceptable? –  skullpatrol Aug 18 '12 at 21:30
@skullpatrol: Probably acceptable although unusual. There is a lack of symmetry in common usage, as far as distinguishing containment of an element vs. containing a subset. On the one hand we may well stress the distinction by saying "$x$ belongs to $G$, but never (that I can recall) that "$G$ possesses (or owns) $x$". Of course you've made the meaning quite clear by adding the phrase "as an element". –  hardmath Aug 19 '12 at 1:05
@BrianM.Scott: Rather than the case of being trapped in my own sentence (however likely that might be!), I was thinking about the case where I needed to verbalize a formula that came from a book or prepared slides, something that might have been elegant enough in print but poses a slight difficulty to "pronounce" as the OP puts it. –  hardmath Aug 19 '12 at 1:08
@hardmath: I’d paraphrase to avoid the problem. –  Brian M. Scott Aug 19 '12 at 12:51

It is very ideosyncratic, but a text in which this is completely explicit is Theory of Value by Gerard Debreu, a classic in mathematical economics. I quote:

Corresponding to these two different concepts, two different symbols, $\subset$ and $\in$, and two different locutions, "is contained in" and "belongs to," are used. Two different verbs are therefore used here to read $\supset$ and $\ni$: for the former "contains," and for the latter "owns," the natural counterpart of "belongs to."

Needless to say, I have never seen or heared "owns" been used in this way somewhere else.

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It may be idiosyncratic, but I am always interested to learn this kind of detail. Thanks very much. –  MJD Aug 29 '12 at 1:58
I like this idea of 'owns'! –  user50229 Jan 3 '13 at 0:41

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