# Relational Division and SQL

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1 Relational Division and SQL Soulé 1 Example Relations and Queries As a motivating example, consider the following two relations: Taken(,Course) which contains the courses that each student has completed, and Required(Course), which contains the courses that are required to graduate. The instances for this example are shown below: Taken Course Databases Programming Languages Susie Databases Susie Operating Systems Julie Programming Languages Julie Machine Learning Operating Systems Required Course Databases Programming Languages Suppose we are asked the following two queries: 1. Find all students who have taken a required course. 2. Find all students who can graduate (i.e., who have taken all required courses). 2 Asking About Some We have already seen how to do Query 1 using the standard SELECT... FROM... WHERE clauses. To remove duplicates from our result, we can use 1

2 the SQL keyword DISTINCT. In terms of relational algebra, we use a selection (σ), to filter rows with the appropriate predicate, and a projection (π) to get the desired columns. SELECT DISTINCT WHERE Course = Databases or Course = Programming Languages ; If we want to be slightly more general, we can use a sub-query: SELECT DISTINCT WHERE Course IN (SELECT Course FROM Required); Informally, we might read this query as give me the set of students who have taken a course that appears in the set of required courses, or the set of students whose courses contain at least one course that is required. 3 Asking About All Query 2 is more difficult. Just by looking at this small instance, it is easy to see that the answer we want is: CanGraduate There is a relational operator that directly gives us this result. The operator is division, written R S. Unfortunately, there is no direct way to express division in SQL. We can write this query, but to do so, we will have to express our query through double negation and existential quantifiers. We will produce this query in stages. Our roadmap is as follows 1. Find all students 2. Find all students and the courses required to graduate 3. Find all students and the required courses that they have not taken 4. Find all students who can not graduate 5. Find all students who can graduate 2

3 All s First, we create a set of all student that have taken courses. We can express this positively using a selection and projection: CREATE TEMP TABLE All as SELECT ; All Susie Julie All s and Required Classes Next, we will create a set of students and the courses they need to graduate. We can express this as a Cartesian product, creating the pairs of the form (student,course): CREATE TABLE sandrequired AS SELECT DISTINCT All.student, Required.course FROM All, Required ; sandrequired Course Databases Programming Languages Susie Databases Susie Programming Languages Julie Databases Julie Programming Languages Databases Programming Languages All s and Required Classes Not Taken This is where our query starts to get tricky. We want to find the subset of the relation we just produced that includes the students and the required courses that they have not taken. We are doing this as a first step towards finding the students who cannot graduate. The intuition is that we want to find all (student,course) pairs that are in the relation sandrequired, but not in the relation Taken. This should give us the set of students who cannot graduate, with the courses that they still need to take: 3

5 Re-write As a Single Query We can re-write the above set of queries into a single query that does not use the temporary tables: SELECT DISTINCT x. FROM taken AS x WHERE NOT EXISTS ( SELECT * FROM required AS y WHERE NOT EXISTS ( SELECT * FROM taken AS z WHERE (z.=x.) AND (z.course=y.course))); Re-write Using Except (or Minus) EXCEPT instead of NOT EXISTS: We can re-write the above to use SELECT EXCEPT SELECT FROM ( SELECT,Course FROM (select ), Required EXCEPT SELECT,Course ); Re-write Using Aggregations We can write the query in an (arguably) simpler version, using set membership (i.e., IN), GROUP BY, COUNT aggregations, and HAVING: SELECT From Taken WHERE Course IN (SELECT Course FROM Required) GROUP BY HAVING COUNT(*) = (SELECT COUNT(*) FROM Required); 5

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