Good point, but not exactly correct. If R is a relation on A x B and S is a relation on B x C then SR = {(a, c) | aRb and bSc for some b in B}. The join uses (a, b, c) in the comprehension.

True, but I think slightly pedantic for this context. I think there have been proposed relational operators which produce the result you describe - a join, then remove the join key fields from the result.

If we are in a pedantic mood, also, a relational-tuple is not exactly the same as a mathematical tuple, relational tuples take different forms depending on who's system you are following (e.g. whether the fields are ordered or not).

Just endeavoring to make true statements. A relation on sets A and B is a subset R of the Cartesian product of A and B. E. F. Codd had something else in mind with his definition of join. Do you think he got that wrong?