Relational Algebra - Cartesian Product vs Natural Join?

Question

I am studying for exams and am failing to find a solid criteria by which I can determine if the Cartesian Product x is to be used or if Natural Join |X|

Answer 1

Cartesian product is just a special case of natural join where the joined relations don't have any attribute names in common. In Codd's original algebra renaming was a separate operation altogether. To obtain a true cartesian product of two relations that have some attributes in common you would have to rename those attributes before doing the (natural) join.

In the interests of brevity the renamings are sometimes left out of written examples and the product symbol used instead. Unfortunately this obscures the important point, that there is only one kind of join.