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I've seen copious usage of prepositions like "on" and "over" in mathematical texts with no concrete description of what they mean. Can someone please precisely define these terms for me, as in, what does it really mean when you say "let x be a y on z?

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Context dependent. Probably more a feature of the English language's variable meanings for many prepositions. –  Michael Joyce Mar 21 '13 at 23:49
    
Usually the denotation is clear. Did you encounter an instance where it was not clear? –  Math Gems Mar 21 '13 at 23:53
    
You can see "Let $R$ be a relation on a set", "Let $V$ be a vector space over a field $F$". The context usually makes it clear what the preposition intends to denote. –  Pedro Tamaroff Mar 21 '13 at 23:57
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It'd be helpful if your request for clarity on the meaning of "on" and "over", which seems to you to lack concrete descriptions, included some concrete examples. –  KCd Mar 21 '13 at 23:57

1 Answer 1

My general impression is that a foo on $X$ is some kind of function, loosely speaking, with domain $X$ while a foo over $X$ is some kind of function, loosely speaking, with codomain $X$. But these terms don't really have completely precise meanings; you learn how to use them from seeing how other people use them (the same way you learned how to use most of the words you know).

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