Help writing the following query to relational algebra

DISCLAIMER : Never Learned Relational Algebra but it looks interesting

From the schema given and your question, this is what the SQL should be:

SELECT
    emp_mgr.person_name
FROM
    manages emp_mgr
    INNER JOIN employee emp ON emp_mgr.person_name  = emp.person_name
    INNER JOIN employee mgr ON emp_mgr.manager_name = mgr.person_name
WHERE
    emp.street = mgr.street AND
    emp.city = mgr.city
;

Here is another query that only uses JOINs, no WHERE clause:

SELECT
    emp.person_name
FROM
    (SELECT A.person_name,B.street,B.city FROM manages A
    INNER JOIN employee B ON A.person_name = B.person_name) emp
    NATURAL JOIN
    (SELECT A.manager_name,B.street,B.city FROM manages A
    INNER JOIN employee B ON A.manager_name = B.person_name) mgr
;

The first query gets all employees who are managed and their managers in the form of a Cartesian Product. Then, it looks for a common street and city.

The second query collects personnel records (name,street,city) of employees and their managers and performs a NATURAL JOIN between the employess and their managers using (street,city).

If you can transalate both queries back to Relational Algebra, I think you will have what you are looking for. I believe the second may be of better help.

Your sql query is wrong:

- Assume there is a student s1 who has passed one exam 
  after '2000-01-01' and none before. 
  Your query results in {s1} - {} = {s1}. This will be a false positive.

- Assume there is a student s1 that passed three exams 
  after '2000-01-01' and one exam before. 
  Your query results in {s1} - {s1} = {}. This will be a false negative.

I'm having a bit of trouble reading your algebra expressions, but the first one looks ok. It should not matter that you do the selection (topic = "motorcycle") after the join instead of joining on the selection.

The second one can't be right. Assume there's a newspaper that published both an article on motorcycle and an article on something else. Your expression will pick the article on something else and therefore return that newspaper (incorrectly).