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In a recent discussion, someone told me tuples in the modern meaning (in particular, tuples are heterogeneous: that is, different elements of a tuple can belong to different sets/have different types) first appeared in Codd's tuple calculus. I was surprised it would be so late, but searching Google Books before 1970, I can't see any clearly heterogeneous examples, and quite a few clearly homogeneous ones ("tuple of ones and zeros", "tuple of natural numbers", etc.)

Сan anybody confirm that Codd introduced heterogeneous tuples or point out an earlier appearance?

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As far as I'm aware, "tuple" is just the genealization of ordered pairs, triples, quadruples, etc. Can examples of heterogeneous pairs and triples really be that new? It seems to me that it must date back at least to the foundations of abstract group/ring theory, e.g. in the formulation of the chinese remainder theorem. –  Niel de Beaudrap Aug 24 '10 at 8:14
WHat was the "earlier" meaning your title implies? –  Mariano Suárez-Alvarez Aug 25 '10 at 19:10
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2 Answers

up vote 5 down vote accepted

Tuples appear in essentially all formal treatments of set theory, and those go back to way before the 70s!

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D'oh. Of course, any definition of a product of different sets does require heterogenous tuples. –  Alexey Romanov Aug 25 '10 at 21:51
And there is a fine example in, e.g., Jacobson's Lectures in Abstract Algebra. –  Alexey Romanov Aug 25 '10 at 21:56
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I'm not sure having really understood your question, but it seems to me that mathematicians make no difference between heterogeneous and homogeneous tuples. Consider $\left(5,b,f,8,3\right)$: one can say it is an heterogeneous tuple because the set $\left\{5,b,f,8,3\right\}$ can be partitioned into $\left\{b,f\right\}$ and $\left\{5,8,3\right\}$, letters and digits respectively. But are letters and digits of different type? What do you mean by type? In mathematics, as far as I know, there is only one type: the set type. Everything is a set. In this example letters and digits should be both defined as (particular) sets. So every tuple is homogeneous by default. Also, consider $\left(2,1,9,7\right)$: there are only digits this time, so one can say it is an homogeneous tuple. But what if I split the set of all digits into those less then $5$ (the "low" digit type) and those equal to or greater than $5$ (the "high" digit type)? It becomes heterogeneous. I would say the context is of great importance here.

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Technically, of course, you are right: everything is a set. But I mean things like "consider a pair ($n$, $a$), where $n$ is a natural number and $a$ is a letter", or, to use your second example, "where $n$ is a number less than 5 and $a$ is a number equal or greater than 5" would be as good; i.e. where the author explicitly says that different elements should belong to different sets. –  Alexey Romanov Aug 25 '10 at 21:55
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