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For the mathematically inclined fellows:

If f is a function whose domain is the set A, do you say that f is defined on A or over A? Do both prepositions apply here or is the use of one of them decidedly wrong?

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As a pure mathematician (albeit a junior one), I'd say that I've only ever seen functions described as being defined on some domain; if you look a bit more carefully at the examples for "defined over" provided in the answer below, you'll find that almost all of them actually refer to functions respecting some sort of structure on the domain and codomain. –  Branimir Ćaćić Jun 22 '13 at 5:00
    
These are not phrasal verbs. The verb here is define. On vs. over are prepositions (not propositions). I have retagged the question accordingly. –  RegDwigнt Jun 22 '13 at 11:00

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up vote 5 down vote accepted

Generally, for mere functions I think defined on A is more common. A quick Google Books search* revealed examples of "defined over A", but that it's more common for a vector space or some more complex algebraic structure defined over A.

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Yes, semantically on is merely 2-D location, but over implies a 3rd dimension with some locational variance. So on the hill simply identifies the hill with the location, while over the hill means either 'located on the other side of the hill' (Path variant) or 'located above the hill' (Flying Saucer variant). The most common mathematical uses I can recall all have some structure, if only an ordering relation, as in defined over R. –  John Lawler Jun 22 '13 at 13:13
    
Not only more common but absolutely normative and obligatory, at least in my (not particularly great) experience. As you say, "defined over A" refers to respecting a certain type of structure. –  Branimir Ćaćić Jun 22 '13 at 16:59

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