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?
asked Jun 22, 2013 at 3:48
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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 ĆaćićCommented Jun 22, 2013 at 5:00
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These are not phrasal verbs. The verb here is define. On vs. over are prepositions (not propositions). I have retagged the question accordingly.– RegDwigнtCommented Jun 22, 2013 at 11:00
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There is a Wikiipedia article which seems to indicate that the prepositions should be used in different cases.– Edwin AshworthCommented Jun 11, 2019 at 16:15
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1 Answer
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. Commented Jun 22, 2013 at 13:13
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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. Commented Jun 22, 2013 at 16:59
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@JohnLawler I like your argument, but the conclusion is to use "over a set", because it is not a single element of the set but you "go over" the entire set, while defining the function.– Make42Commented Jun 22, 2020 at 15:48