# Let Ω be a domain in/of Rⁿ

What is the correct preposition in the sentence:

Let Ω be a domain in/of Rⁿ.

Is there a different meaning for "in" and "of"? Both seem to be commonly used, Google gives about 200.000 hits for both, "domain in RN" and "domain of RN".

(Disclaimer: I'm a non-native speaker)

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This might have been asked at Mathematics. Since you've identified yourself as a non-native speaker, I'll also pay a brief mention to English Language Learners, too – although I wouldn't suggest migrating this particular question there. – J.R. Sep 16 '13 at 9:42
This question appears to be off-topic because it is about Mathematics and its terminilogy. – Canis Lupus Sep 16 '13 at 13:21
Not Mathematics. This questions is about the English language. The answer, though, requires an understanding of the mathematical context. – Kris Sep 17 '13 at 7:24

Presumably the sentence uses domain in a sense applicable in mathematical analysis, not as related to functions or ring theory. Per wikipedia,

In mathematical analysis, a domain is any connected open subset of a finite-dimensional vector space. This is a different concept than the domain of a function...

The appropriate preposition is in; the sentence “Ω is a domain in Rⁿ” asserts that Ω is a connected open subset of Rⁿ, which is an n-dimensional vector space. One also can say “Ω is a set in Rⁿ” and “Ω is a subset of Rⁿ”. Using * to mark incorrect elements, one would not say any of “Ω is a domain *of Rⁿ”, “Ω is a set *of Rⁿ” and “Ω is a subset *in Rⁿ”, although the third of those is less egregiously wrong than the other two.

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Domain "of" would imply that Ω is the starting set of numbers from which all of the values for a function RN are calculated. This is the most correct usage in terms of English and Mathematics.

Domain "in" would imply that RN is somehow biggest... and entirely contains the domain Ω. However, this is not a common usage, because a 'domain' is generally the set of numbers that represents the inputs for a function.

With your short context, I would assume that you want Let Ω be the domain of RN. This would represent that set of numbers Ω that should be fed into function RN.

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Your reasoning is not bad, but is based on misunderstanding what R^n is. R is the set of real numbers, R^n is the set of n-tuples whose members are all real numbers. – Ben Voigt Sep 16 '13 at 18:26
Check my comment at OP. – Kris Sep 17 '13 at 7:25

It's unclear whether you are talking about the domain of a function which is a subset of Rn or whether you are using a more technical use of the word domain. Even if your sentence is technically grammatically correct, it needs context to be clear. If it is the usual meaning (such as in 'the domain of the square root function is the nonnegative reals') then the correct use is 'Let Ω, a subset of Rn, be the domain of a function...'

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I think in some cases they are interchangeable.

In some other not. For instance see this question:

"Why any function with domain in R can be written..." http://answers.yahoo.com/question/index?qid=20121027120143AA48ZPa

So if you are talking of the "domain of a function", there seems to be only one choice: the domain is "in".

If you, instead, mean a subset of R, I would guess there is not much difference: I would take a look at the most authoritative native authors and follow their example.

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