Which students took every Math course? A join finds "at least one"; it cannot say "every". Divisiondivisionr ÷ s: the tuples t over the attributes of r that are not in s, such that t combined with every tuple of s is in r can.

Definition

r(R) ÷ s(S), where every attribute of S is also in R:

• the schema of the result is R − S (the attributes of r that are not in s),
• a tuple t is in the result when, for every tuple u in s, the tuple (t, u) is in r.

If s is empty, every t in π R−S (r) passes, because there is no u to miss.

Division is a short form of the fundamental operators:

r ÷ s  =  π R−S (r)  −  π R−S ( (π R−S (r) × s)  −  r )

Read the right side from the inside: π R−S (r) × s is every pair that "should" exist. Subtract r: the pairs that are missing. Their t values fail. Remove them from all t values.

In SQL

SQL has no ÷. Write "for every" as "no course is missing":

SELECT DISTINCT t.id
FROM takes AS t
WHERE NOT EXISTS (
  SELECT course_id FROM course WHERE dept_name = 'Math'
  EXCEPT
  SELECT t2.course_id FROM takes AS t2 WHERE t2.id = t.id
);

The inner query is "the Math courses that this student did not take". NOT EXISTS keeps the student when that is empty.