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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.

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    $\begingroup$ "On" or "in" is fine and interchangeable here. "Replaces" wouldn't be quite right, since when we replace something one thing has to go away: neither $u$ nor $v$ disappear here. $\endgroup$ – Randall Oct 6 '17 at 13:41
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    $\begingroup$ An operation is done on a given set, even if it takes place in the given set. Example: A group $G$ acts on a set $X$, not in a set $X$. $\endgroup$ – Dietrich Burde Oct 6 '17 at 13:43
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    $\begingroup$ Usually, in math, we use the term operation: "an operator is generally a mapping (a function) that acts on the elements of a space (a set $X$) to produce other elements of the same space ($X$)." Thus, addition is an operation defined on a set $V$ that assigns to every pair $u,v$ of elements of $V$ an element of $V$. $\endgroup$ – Mauro ALLEGRANZA Oct 6 '17 at 13:44
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    $\begingroup$ I think "on" is more correct since "addition" is a particular type of "function", and on usually speaks of a "function on a set" rather than a "function in a set". Also, "assign" is the action of the function, so that's normal as well; it indicates $(u,v)\mapsto u+v$. $\endgroup$ – MPW Oct 6 '17 at 13:45
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    $\begingroup$ An employer can assign a job to an employee: that person is associated with the job. If the employer replaces the employee with someone else, then he or she is gone! $\endgroup$ – Théophile Oct 6 '17 at 14:24
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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.

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