What's the difference between the Cartesian product in set theory and in relational algebra?

The Cartesian product in set theory is defined as:

$$A \times B = \{(a, b) \mid (a \in A) \land (b \in B)\}$$

I think this is exactly how it works in relational databases, but Wikipedia tries to make a difference that I don't understand:

$$ R \times S := \{ (r_1,r_2,\dots,r_n,s_1,s_2,\dots,s_m) \mid (r_1,r_2,\dots,r_n) \in R, (s_1,s_2,\dots,s_m) \in S \} $$

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    $\begingroup$ Why do you think those are the same? Why not just work through an example? A & B are sets of values, but R & S are sets of tuples of values. $\endgroup$ – philipxy May 6 '17 at 6:36
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    $\begingroup$ Relations in the relational algebra are not the ordered-tuple relations of math & "binary relations". It is a different theory. RA-style relations involve sets of n-ary tuples; frequently the tuples are unordered with tagged elements, and usually the sets are paired with headings. $\endgroup$ – philipxy May 6 '17 at 6:40

The Wikipedia definition merely "opens the parentheses". Using the pure set-theoretic definition, we would expect $$ ((r_1,r_2,\dots,r_n),(s_1,s_2,\dots,s_m)) $$ instead of $$ (r_1,r_2,\dots,r_n,s_1,s_2,\dots,s_m). $$ Assuming $R,S$ are both relations over the same universe $U$, the Wikipedia definition ensures that $R \times S$ is an $(n+m)$-ary relation over $U$. Under the set-theoretic definition, we get a binary relation over $U^n$ when $n = m$, and a unary relation over $U^n \times U^m$ in general.

We often write $U^n \times U^m = U^{n+m}$, but this isn't in general an equality of sets (it might be in some special cases, depending on the exact definition of $U^n$). Rather, we identify $U^n \times U^m$ with $U^{n+m}$, that is, we ignore the difference (this informal notion can be formalized). The difference between $R \times S$ as sets and as relations is similar – we identify the two objects although they are "physically" different.

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  • $\begingroup$ Thank you very much for your answer Yuval. I edited my question: the link to Wikipedia was wrong. I'm just learning about binary relations and I thought they are possible even when n != m (If I understand correctly and n and m are the cardinals of R and S). For example I thought this was correct: A = {2, 4} and B = {1, 2, 3, 4, 5} A binary relation "less than", R = {(2,3), (2,4), (2,5), (4,5)} Although I know about the universal set U, I get a bit lost when I read U^n So please correct me if my understanding about binary relations is wrong. $\endgroup$ – Víctor May 5 '17 at 21:58
  • $\begingroup$ Yes, it depends on the definition of relation. In many contexts we want a n $n$-ary relation to be a subset of $U^n$ for some set $U$, whereas is other contexts this is not so important. $\endgroup$ – Yuval Filmus May 5 '17 at 22:02

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