1. The Relational Model
L2DB

2. Agenda The Relational Algebra Operations Expressions
Tuple Relational Calculus
Domain Relational Calculus
Implications for Query Processing
Summary

3. Relational Algebra operations
Restrict r
Project p
Cartesian product ´
Natural join
Union 
Intersection 
Difference -
others

4. An example schema Book book# title author classCode publisher year
Member
member#
surname
forename
tel#
address
Loan
book#
member#
dueDate
Classification
classCode
className

5. Restrict r
The Restrict operation selects a subset of tuples from the input relation according to some condition
Specified by r<selection-condition> (R)
Examples:
rauthor=‘Dickens Charles’ (Books)
rdueDate<= date(‘today’) (R)

6. Restrict in action rtel#=1131 Member member# surname forename tel#
address
Shah
Hanif
4123
1002
30 High St.
Piaf
Therese
4178
1131
11, The Crest
Thom
Natalie
5016
1132
11, The Crest
Johns
Phil
4268
1098
77, The Lane
O’Reilly
Sean
5286
1131
11, The Crest
rtel#=1131
anon
member#
surname
forename
tel#
address
Piaf
Therese
4178
1131
11, The Crest
O’Reilly
Sean
5286
1131
11, The Crest

7. Project p
The Project operation returns a subset of attributes from the input relation
Specified by p<attribute-list> (R)
Examples:
p author,title (Books)
p book# (R)

8. Project in action Ptel#,surname Member member# surname forename tel#
address
4123
Shah
Hanif
1002
30 High St.
4178
Piaf
Therese
1131
11, The Crest
5016
Thom
Natalie
1132
11, The Crest
4268
Johns
Phil
1098
77, The Lane
5286
O’Reilly
Sean
1131
11, The Crest
Ptel#,surname
anon
tel#
surname
1002
Shah
1131
Piaf
1132
Thom
1098
Johns
1131
O’Reilly

9. Cartesian Product ´
The Product operator takes in two relations and produces an output in which every tuple in the first relation is matched with every row in the second relation
Specified by
R1 ´ R2
Note: The operator takes its name from Rene Descartes, the French philosopher mathematician

10. Product in action ´ Loan Member book# member# dueDate member# surname
forename
tel#
address
57862
4178
22 Sept 08
4123
Shah
Hanif
1002
30 High St.
31991
4123
15 Oct 08
4178
Piaf
Therese
1131
11, The Crest
22264
4178
02 Sept 08
anon
book#
member#
dueDate
member#
surname
forename
tel#
address
57862
4178
22 Sept 08
4123
Shah
Hanif
1002
30 High St.
57862
4178
22 Sept 08
4178
Piaf
Therese
1131
11, The Crest
31991
4123
15 Oct 08
4123
Shah
Hanif
1002
30 High St.
31991
4123
15 Oct 08
4178
Piaf
Therese
1131
11, The Crest
22264
4178
02 Sept 08
4123
Shah
Hanif
1002
30 High St.
22264
4178
02 Sept 08
4178
Piaf
Therese
1131
11, The Crest

11. Natural join
Natural join is the most common form of Join operation. It combines the data from two relations by matching values on attributes with the same name.
Specified by
R R2

12. Natural join in action
Loan
Member
book#
member#
dueDate
member#
surname
forename
tel#
address
57862
4178
22 Sept 08
4123
Shah
Hanif
1002
30 High St.
31991
4123
15 Oct 08
4178
Piaf
Therese
1131
11, The Crest
22264
4178
02 Sept 08
anon
book#
member#
dueDate
surname
forename
tel#
address
57862
4178
22 Sept 08
Piaf
Therese
1131
11, The Crest
31991
4123
15 Oct 08
Shah
Hanif
1002
30 High St.
22264
4178
02 Sept 08
Piaf
Therese
1131
11, The Crest
Note: the effect of a natural join can be achieved by a Cartesian Product
followed by a Restrict and a Project - see if you can work out how.

13. Union 
Union takes in two compatible relations and returns a relation that contains the tuples that appear in either or both.
The relational algebra demands that the input relations are compatible, i.e. they are of the same type

14. Union in action  StaffLoan StudentLoan book# member# dueDate book#
57862
4178
22 Sept 08
31991
4123
15 Oct 08 
31991
4123
15 Oct 08
32277
4178
18 Sept 08
22264
4178
02 Sept 08
59999
4690
29 Oct 08
59999
4691
29 Oct 08
anon
book#
member#
dueDate
57862
4178
22 Sept 08
31991
4123
15 Oct 08
22264
4178
02 Sept 08
32277
4178
18 Sept 08
59999
4690
29 Oct 08
59999
4691
29 Oct 08

15. Intersection 
Intersection takes in two compatible relations and returns a relation that contains the tuples that appear in both.
Again the relational algebra demands that the input relations are compatible, i.e. they are of the same type

16. Intersection in action
StaffLoan
StudentLoan
book#
member#
dueDate
book#
member#
dueDate
57862
4178
22 Sept 08
31991
4123
15 Oct 08 
31991
4123
15 Oct 08
32277
4178
18 Sept 08
22264
4178
02 Sept 08
59999
4690
29 Oct 08
59999
4691
29 Oct 08
anon
book#
member#
dueDate
31991
4123
15 Oct 08

17. Difference -
The difference operator takes in two compatible relations X and Y and returns a relation that contains the tuples that appear in X but NOT in Y

18. - Difference in action StaffLoan StudentLoan book# member# dueDate
57862
4178
22 Sept 08
31991
4123
15 Oct 08
31991
4123
15 Oct 08
32277
4178
18 Sept 08
22264
4178
02 Sept 08
59999
4690
29 Oct 08
59999
4691
29 Oct 08
anon
book#
member#
dueDate
57862
4178
22 Sept 08
22264
4178
02 Sept 08

19. Other operators Codd originally described a set of eight operators
Some of these operators are not essential because they can be derived from others
Many authors have contributed further operators to the algebra
divide, semijoin, outer join, …
The set of operators provided in any relational implementation is relationally complete if it is at least as powerful as Codd’s original set.

20. Relational Algebra expressions
The power of relational algebra arises from its ability to compose expressions
Examples
p title (rauthor=‘Greene Graham’ (Book))
will return the titles of books written by Graham Greene
p title,author(rmember#=4128 (Loan Book))
will return the titles and authors of books on loan to library member 4128

21. Relational Algebra trees
p title((rmember#=4128 Loan) Book) Member )
An expression, such as the above can be represented as a tree:
p title
Expression
result
Intermediate
results
rmember#=4128
Loan Book Member
Base relvars

22. Equivalence of algebra expressions
Consider the following two expressions
p title,surname((rmember#=4128 Loan) Book) Member )
ptitle,surname(rmember#=4128 ((Loan Book)) (psurname,member#Member )))
The expressions are quite different in the operations, particularly in the
order in which the operations are applied.
However, the two expressions will always yield the same result i.e. the
expressions are equivalent.
If we consider, the size of the intermediate results and the number of
tuples processed by each operator, we can assess that the first expression
is likely to be more efficient than the second.

23. Implications for Query Processing
The database query language SQL is (loosely) based on the tuple relational calculus
The database query language QBE is based on the domain relational calculus
Either of these can be converted into relational algebra, which provides a processing model to execute the query
In a relational DBMS a software component known as a Query Optimizer converts a query into an efficient operation tree that is similar to the relational algebra

24. Summary The relational model comprises: The model
a structural specification
a means of enforcing integrity
a means of manipulating data
The model
provides a solid theoretical basis for database processing, based on set theory and first-order predicate logic
supplies the foundation for relational DBMS implementation

25. Caveat
Commercial RDBMS products and the SQL standard language are based on the relational model
However, none of them conform to the relational model
the extent to which existing systems deviate from the model and the implications of this is a matter of great controversy
see the reading list for further study of this area