SQL Query INTERSECTS - Relational Algebra - mysql

Question

I am trying to convert the following 2 relational algebra into standard english but I am having trouble understand what exactly the INTERSECTS does
I know that INTERSECT finds the values that occur in both select statements
I have managed to find its SQL query, but I just can't figure out what it asks in plain english
Would anyone be able to help?
Relational Algebra:
Πstudent_name(σcourse id=“CS1850“(student Natural Join enrolled)) ∩
Πstudent_name(σcourse id=“CS1840“(student Natural Join enrolled))
SQL:
SELECT student_name FROM student NATURAL JOIN enrolled WHERE
course_id='CS1850' INTERSECT SELECT student_name FROM student
NATURAL JOIN enrolled where student_name='CS1840'
Second Expression:
Πstudent_name(σe1.student
id=e2.student_id∧e1.course_id<>e2.course_id(student natural join
(ρe1(enrolled) × ρe2(enrolled))))
SELECT student_name FROM student NATURAL JOIN enrolled e1 NATURAL JOIN
e2 WHERE e1.student_id = e2.student_id

Answer 1

First, don't use NATURAL JOIN. It is an error waiting to happen. It matches tables based on the names of columns in common. It doesn't even use properly declared foreign key relationships. Instead, use USING or ON to be explicit about the columns used for joining.
Second, you don't need INTERSECT. There are two typical methods that use GROUP BY or JOIN.
In English, the query gets the students that are enrolled in both classes.
SELECT s.student_name
FROM student s JOIN
enrolled e
USING (student_id) -- or whatever
WHERE e.course_id IN ('CS1850', 'CS1840')
GROUP BY s.student_name
HAVING COUNT(*) = 2;
Note: This assumes that two students do not have the same name.
An alternative just uses JOIN:
SELECT s.student_name
FROM student s JOIN
enrolled e1
USING (student_id) JOIN
enrolled e2
USING (student_id)
WHERE e1.course_id = 'CS1850' AND
e2.course_id = 'CS1840';
If the underlying tables do not have duplicates, then this doesn't need a GROUP BY or DISTINCT.