Difference between the Cartesian product in set theory and in relational algebra

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:

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

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.

StackExchange Computer Science Q#74970, answer score: 2