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# Exercise: Relational Algebra

Module by: Nguyen Kim Anh. E-mail the author

## Exercise 1

Consider the relations:

 r(A B C) s(D E F) t(A B D) a1 b1 1 d1 e1 1 a1 b1 d1 a2 b2 4 d2 e2 4 a2 b2 d2 d3 e2 8 a1 b3 d3 d3 e3 1 a2 b1 d2 a3 b3 d3 a3 b1 d1 a2 b3 d3

Show the results of the following relational algebra expressions

1. Π AB ( σ D <> d1 ( t ) ) AB ( r ) Π AB ( σ D <> d1 ( t ) ) AB ( r ) size 12{Π rSub { size 8{ ital "AB"} } $$σ rSub { size 8{D"<>"d1} } \( t$$ \) \Π rSub { size 8{ ital "AB"} } $$r$$ } {}
2. r * t * s
3. t ÷ Π A ( r ) * s t ÷ Π A ( r ) * s size 12{ left (t div Π rSub { size 8{A} } $$r$$ right ) * s} {}
4. r C F s r C F s size 12{r ⊳ ⊲ rSub { size 8{C >= F} } s} {}

## Exercise 2

Consider the relations

 r(A B C) s(D E ) t(A D G) a1 b1 1 d1 e1 a1 d1 11 a2 b2 2 d2 e2 a1 d2 12 a3 b3 3 d3 e3 a2 d1 21 a3 b1 4 a2 d2 22 a3 d1 31 a2 d3 23

Show the results of the following expressions.

• Π EG ( σ A = a2 A = a3 ( s t ) ) Π EG ( σ A = a2 A = a3 ( s t ) ) size 12{Π rSub { size 8{ ital "EG"} } $$σ rSub { size 8{A=a2 or A=a3} } \( s*t$$ \) } {}
• Π AD ( t ) ÷ Π A ( r ) Π AD ( t ) ÷ Π A ( r ) size 12{Π rSub { size 8{ ital "AD"} } $$t$$  div `Π rSub { size 8{A} } $$r$$ } {}

## Exercise 3

Consider two relations R1 and R2, where R1 contains N1 tuples and R2 contains N2 tuples, and N1 > N2 > 0. Give the minimum and maximum possible sizes (in tuples) for the result relation produced by each of the following relational algebra expressions. In each case state any assumptions about the schemas of R1 and R2 that are needed to make the expression meaningful.

• R1 size 12{ union } {} R2
• R1 size 12{ intersection } {} R2
• R1 \ R2
• R1 × R2
• σ a = 5 ( R1 ) σ a = 5 ( R1 ) size 12{σ rSub { size 8{a=5} } $$R1$$ } {}
• π a ( R1 ) π a ( R1 ) size 12{π rSub { size 8{a} } $$R1$$ } {}
• R1 size 12{ div } {} R2

## Exercise 4

Consider the following relational schema:

Parts(pid, pname, colour)

Catalog(supplier, part, cost)

Assume that the ids are integers, that cost is a real number, that all other attributes are strings, that the supplier field is a foreign key containing a supplier id, and that the part field is a foreign key containing a part id. Write a relational algebra expression to answer each of the following queries:

• Find the names of suppliers who supply some red part.
• Find the sids of suppliers who supply some red or green part.
• Find the sids of suppliers who supply some red part or whose address is 221 Packer Street.
• Find the sids of suppliers who supply some red part and some green part.
• Find the sids of suppliers who supply every part.
• Find the sids of suppliers who supply every red part.
• Find the sids of suppliers who supply every red or green part.
• Find the sids of suppliers who supply every red part or supply every green part.
• Find the pids of parts that are supplied by at least two different suppliers.
• Find pairs of sids such that the supplier with the first sid charges more for some part than the supplier with the second sid.
• Find the pids of the most expensive part(s) supplied by suppliers named Yosemite Sham.

## Exercise 5

Consider the following database schema

Course (Code, Name, Lecturer)

Specify the following queries in relational algebra

1. List the name of all courses
2. List the StudentId, FullName of students who lives in Hanoi
3. List the StudentId of students who enrol in course ‘COMP-1101’ but not in course ‘COMP-1102’
4. List the name of students who enrol in both course ‘COMP-1101’ and course ‘COMP-1102’
5. List the name, lecturer of all course in which student ‘John Smith’ has enroled
6. List the name of student who enrol in at least one course teach by Professor Le Quan
7. List the name, address of all student who enrol in every course teach by Professor Le Quan
8. List the name of the course which has no student enrolled in

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