Mathematics and Grammar I am reading  "Linear Algebra done right" Book (3rd) edition by Axler. I am trying to understand the following statement :
"An addition on a set V is a function that assigns an element u+v ∈ V to each pair of elements u,v ∈ V" 
I am not a native English speaker, therefore, I am confused by the use of the preposition "addition on a set" shouldn't be an addition in a set ? also the word "assigns" seems to me as "replaces".
Thanks in advance for your insights.
 A: One of the lessons that was pounded into me while trying to learn Russian and German (as a native English speaker) is that prepositions are a royal pain in the ass.  In this case, I think that you would be understood if you said either "addition on" or "addition in" $V$.
That being said, there are subtle reasons that "on" might be slightly preferable.  One way of thinking of binary relations (addition, multiplication, etc) is as functions with a domain consisting of the two-fold Cartesian product of the set.  That is, addition is a function
$$ + : V\times V \to V. $$
Addition can be defined on this Cartesian product, thus as a shorthand, it makes sense to say "addition on $V$."
Moreover, as others have pointed out in the comments, we often think of groups acting on other sets.  In a really abstract way, we can think of an underlying groug acting on $V$ to give vector addition.  I wouldn't worry too much about such details—at least, not until you have taken a few courses in modern or abstract algebra—but you might eventually be able to make some sense of it.
All of that being said, if I really wanted to emphasize that a set is closed under an operation, I might be tempted to say that the operation is defined "in" a set.
Long story short:  Either preposition is fine.  There are subtle distinctions to the native ear.  "On" is probably slightly preferable in most situations.
