2. Division
 Division is not essential operation ; just a useful shorthand.
 Also true of joins, but joins are so common that systems implement
joins specially. Division is NOT implemented in SQL.
 Not supported as a primitive operator, but useful for many
expressing queries
 Precondition: in A/B, the attributes in B must be included in the
schema for A. Also, the result has attributes A-B.
 SALES(supId, prodId);
 PRODUCTS(prodId);
 Relations SALES and PRODUCTS must be built using projections.
 SALES/PRODUCTS: the ids of the suppliers supplying ALL
products

3.  The division is a binary operation that is written
as R ÷ S
 The result consists of the restrictions of tuples in R to
the attribute names unique to R
 The header of R but not in the header of S, for which it
holds that all their combinations with tuples in S are
present in R

4. Student Task
Fred Database1
Fred Database2
Fred Compiler1
Eugene Database1
Eugene Compiler1
Sarah Database1
Sarah Database2
Completed DBProject
Task
Database1
Database2
Completed
÷
DBProject
Student
Fred
Sarah

5. sNo pNo
s1 p1
s1 p2
s1 p3
s1 p4
s2 p1
s2 p2
s3 p2
s4 p2
s4 p4
pNo
p2
A B1 A/B1
sNo
s1
s2
s3
s4

6. sNo pNo
s1 p1
s1 p2
s1 p3
s1 p4
s2 p1
s2 p2
s3 p2
s4 p2
s4 p4
pNo
p2
p4
A B2 A/B2
sNo
s1
s4

7. sNo pNo
s1 p1
s1 p2
s1 p3
s1 p4
s2 p1
s2 p2
s3 p2
s4 p2
s4 p4
pNo
p1
p2
p4
A B3 A/B3
sNo
s1

8.  Suppose that employees of the company are assigned
to some professional courses; each employee follows
several courses and each course is taken by several
employees
 Information about courses and which employee takes
which course can be represented by the following
relations
 Takes (Emp#, Course#)
 Courses (Course#, Topic, Weeks)

9.  Assume that these two relations have the following
content
Emp# Course#
7369 10
7369 20
7369 30
7782 10
7782 40
Course# Topics Weeks
20 CAD 5
10 DB 3
30 DB 2
40 OS 4
Takes Courses

10.  Consider the following query: “determine the emp# of
the employees taking all the courses with topic DB”
(Q1)
 The course# of the courses relevant to the query is
obtained through the following expression:
R1 = ∏ Course# (σ Topic=“DB”) (Courses))
The result of such query is {10, 30}

11.  The result of query Q1 is represented by all the
employees that appear in the Takes relation with each
of the course numbers retrieved by query R1
 The correct result of Q1 is thus {7369}
 The operation that allows one to execute such query is
the division

12.  Let R and S be relations; let UR and US be the attribute
 sets of R and S, respectively, and such that UR US
 The division operation is denoted as
R ÷ S
and is expressed as follows
 ∏(UR -US) (R) - ∏ (UR -US) ((∏ (UR –US) (R) X S) – R)
 The expression on the right side of the – sign
determines all tuples of R that are not associated with
at least a tuple of S

13.  Query Q1 is thus expressed as follows:
Takes ÷ ∏ Course# (σ Topic=“DB”) (Courses))
R = Takes
S = ∏ Course# (σ Topic=“DB”) (Courses))
S = {10, 30}
UR = {Emp#, Course#}
Us = {Course#}