I am studying for exams and am failing to find a solid criteria by which I can determine if the Cartesian Product
tableA.colname1 = tableB.colname1
takes(ID, course_id, sec_id, semester, year, grade)
student(ID, name, dept_name, tot_cred)
π name(σ semester="Spring" ^ year=2011(takes ⋈ student)) ∪ π name(σ semester="Autumn" ^ year=2011(takes ⋈ student))
π name(σ semester="Spring" ^ year=2011 ^ takes.ID=student.ID(takes x student)) ∪ π name(σ semester="Autumn" ^ year=2011 ^ takes.ID=student.ID(takes x student))
A natural join, as I understand it, is a projected, filtered Cartesian product:
Under this assumption, your answer is isomorphic to the actual answer.
To see this, you might want to expand the natural join to the above sequence of operators, and float them around using the laws of relational algebra. You'll see that the projection disappears due to the projection to
name, and the selection criterion is fused with the selection above. You'll end up with exactly the same tree as the actual answer, even though you never changed the meaning of your own answer!
I can think of one reason why your lecturer uses these concepts interchangeably: your lecturer wants you to understand that these concepts can be used interchangeably, because "the natural join is just a shortcut" (though that's debatable).