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I've read from the theory of sets that the definition of ordered pair is foundational. To define formally the term of $n$-tuple I suppose we need to use the concept of ordered pair as well as the definition of recursion. So I was wondering about the difference between doing so and defining an $n$-tuple just like we do with matrices, I mean we can just say that an $n$-tuple is a sequence (a function with domain $\{ 1,2,3,4,...,n\}$). What is the advantage of making a definition by recursion? What is the difference between a sequence and an $n$-tuple? What is the difference between considering a vector as a matrix and considering a vector as $n$-tuple?

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There is no difference if you just do ordinary operations, e.g. addition, matrix multiplication etc. There is a difference if you do unusual operations, like union of two vectors (which you can do, because you can define them to be sets). But who would do that? –  Julian Kuelshammer Feb 18 '13 at 15:42
in some cases is used the concept of family... Examples: en.wikipedia.org/wiki/Indexed_family#Examples_2 ;) –  mle Aug 10 '13 at 11:12

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There is no actual difference for the finite case, that is both definitions as tuples and by recursion are just fine. Many people define ordered pairs, then use them to define $n$-tuples as functions from $n$ into the set, and using those even for 2-tuples instead of the original ordered pairs.

The advantage of this approach is in the sort of "universal" approach, but more importantly is that it allows us to define any $\alpha$-tuple for an ordinal $\alpha$, because we cannot recursively define a tuple of infinite length. Using the function definition works, and it works very nicely.

Practically there is absolutely no difference. We merely encode an abstract mathematical construct by sets and there are plenty of ways to do that. This is similar to the decision of saving your database as one structure or another, it may make a difference in the internal processing, but for the end-user it is usually uninteresting and unimportant.

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Just a remark on your statement the definition of ordered pair is foundational.

Do you mean an ordered pair is defined axiomatically? Anyway, a way to define an ordered pair, which I learned from Halmos' Naive Set Theory, is by $$ (a, b) := \{ \{ a \}, \{ a, b \} \}. $$ It has all the properties you desire of an ordered pair, plus some incidental ones you may safely disregard.

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This definition is due to Kuratowski by the way. –  Asaf Karagila Feb 18 '13 at 16:20
@AsafKaragila, thanks. –  Andreas Caranti Feb 18 '13 at 16:21

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