How do you pronounce the inverse of the $\in$ relation? How do you say $G\ni x$?

Question

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.]

Answer 1

Paul Halmos in How to write mathematics 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 $x\in G$.

Answer 2

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

Answer 3

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.