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I just stumbled upon a linguistic problem I wasn't able to resolve via web search. Suppose we're given some geometric set $A$ and subset $B\subset A$. Isn't there a compact way of saying that there exists, say, a plane whose intersection with $A$ is equal to $B$ of the form "there exists a plane intersecting $A$ (preposition) $B$"?

AFAICT, the prepositions "at", "by" and "along" do not fit in general. Maybe I refuse to accept that there is no such construction in English because I'm just too used to the analogous Russian expression.

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    $\begingroup$ I use "in" for this. $\endgroup$
    – Steven Landsburg
    Commented Feb 3, 2014 at 2:36
  • $\begingroup$ Hmm... It seemed somehow off to me before. But now that you've mentioned it, it actually seems to be the one! =) $\endgroup$
    – Igor Makhlin
    Commented Feb 3, 2014 at 2:45
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    $\begingroup$ I'd agree that "in" is correct, but potentially misread. Changing the quoted phrase in the question to "there exists a plane whose intersection with A is B" is unambiguous and scarcely longer than the more ambiguous. $\endgroup$
    – paul garrett
    Commented Feb 3, 2014 at 3:01
  • $\begingroup$ Although I use "in" for this all the time in conversations, I completely agree with @paulgarrett that for formal writing --- where there's no opportunity for the reader to ask for clarification --- it's much better to find a completely unambiguous construction. $\endgroup$
    – Steven Landsburg
    Commented Feb 3, 2014 at 4:05
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    $\begingroup$ It doesn't take too much remodeling to make this unambiguous: there exists a plane whose intersection with A is B. $\endgroup$
    – Vidit Nanda
    Commented Feb 3, 2014 at 4:31

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I think the answer to your question is no---at least modulo any errors in the classification of English prepositions given on wikipedia (which classification includes modulo, by the way). There are a number that almost work and others that might make you smile but none that really do the job right.

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  • $\begingroup$ It's missing a very important preposition because prescriptivism. $\endgroup$
    – Yoav Kallus
    Commented Jan 21, 2015 at 5:33

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