1. Relational Algebra Chapter-2
Prepared By: Muhammad Arshad Javed
Reference: Miss Jenny Coady
Heriot-Watt University

2. Motivation
SQL is a declarative language that allows the user to express what information they require from the database.
It does not allow the user to specify how that information will be extracted. This has the benefit from the user's point of view of making SQL much more usable because there is no need to get involved in the detailed mechanisms involved in satisfying the query.

3. 2.
Because of this, the responsibility for developing procedures for satisfying queries devolves to the Query Optimiser, which forms part of the DBMS. Relational algebra is a mathematical formalism that is used to express queries. It forms the basis of the procedural language of the execution plans, which are generated by the Query Optimiser. Execution plans specify how to satisfy the query. They list a sequence of procedures which can be applied to the database in order to satisfy the query.

4. 3.
The Relational Algebra consists of a set of fundamental operators that take relations as their operands and return a relation as their result. This means that operators can be composed, ie the output of one operation may be used as input to another operation.
The relational algebra is a language of expressions that can be evaluated to yield relations. In a similar way, normal algebraic expressions, for example x+5*y, are evaluated to yield numbers. In normal algebra, one expression can be equivalent to another expression, for example, x+5*y = y*5+x.

5. 4.
The operators defined in the Relational Algebra enable the Query Processor of the DBMS to analyse the structure of queries, to generate alternative strategies for solving them, to evaluate the efficiency of competing strategies, and to manipulate the data in the tables in order to generate query results.

6. Relational Algebra Operations
Basic Operations:
SELECTION, σ - Selects the rows, or tuples, from a relation, which satisfy the specified constraints or restrictions
PROJECTION, π - Selects the specified columns, or attributes, from a relation
CROSS PRODUCT, X - Combines the tuples from two relations to create new tuples containing attributes from both original relations. Every tuple from one relation is combined with every tuple from the other relation.

7. Operations 2.
UNION, U - creates a new relation containing all the tuples from two relations which have the same structure.
SET DIFFERENCE, \ - creates a new relation containing tuples from two relations which have the same structure, and where the tuples exist in the first relation but not in the second relation.

8. Operations 3. Some additional operations are: Intersection “∩ “
Join “ “
Renaming “ρ “
Operations rake relations as input, and output relations
~ operations can be composed!

9. Example Relation
Select is used to select specific tuples, or rows, from a single relation. We use the Greek symbol for s, sigma or σ to represent the SELECT operator.
In SQL, we specify the tuples to be selected in the WHERE clause. For example:
SELECT *
FROM customer c
WHERE c.customer_no < 6 and c.customer_no>= 2
This SQL statement selects only those tuples from the relation, Customer, where the customer number is less than 6 and greater than or equal to 2.

10. Example SELECT
If the original relation, Customer, contained the following data:
customer_no
name
address
tel_no 1
Anne Wetherby
15 Broomilea Drive Denpasar EX1 2XS 2
Peter Smith
22 Clarence Drive Wolverhampton WH3 2DX 3
Ben Hinckley
2 Hexam Drive Hexam HQ12 4TG 4
Robert Harthill
3 Coleshill Birmingham BH1 6GT 5
Peter Michaels
26 Market Square Leicester LE19 3RT 6
Dudley Smith
44 Tamworth Crescent Walsall WA19 6HJ 7
10 PinkLady Drive Kuala Lumpur EX1 2XS

11. EXAMPLE SELECT Then the result of the SELECT is as below
customer_no
name
address
tel_no 2
Peter Smith
22 Clarence Drive Wolverhampton WH3 2DX 3
Ben Hinckley
2 Hexam Drive Hexam HQ12 4TG 4
Robert Harthill
3 Coleshill Birmingham BH1 6GT 5
Peter Michaels
26 Market Square Leicester LE19 3RT
Four rows have been selected from the relation, Customer.
The equivalent relational algebra expression follows:
σc.customer_no>=2 and c.customer_no<6Customer

12. Example Projection Deletes attributes not in the projection list
Schema of result contains the attributes in the projection list
Project is used to select specific attributes, fields, or columns, from a single relation. We use the Greek symbol for p, pi or Π to represent the PROJECT operator.

13. Example PROJECTION
name
tel_no
Anne Wetherby
Peter Smith
Ben Hinckley
Robert Harthill
Peter Michaels
Dudley Smith
In SQL, we specify the attributes to be selected in the SELECT clause. For example:
SELECT c.name, c.tel_no
FROM customer c
Then the result would be
The equivalent relational algebra expression follows:
πc.name, c.tel_noCustomer

14. EXAMPLE JOIN
Join is used to combine two or more relations in order to form, as the result, a new relation, consisting of new tuples containing attributes from all the original relations which satisfy the specified conditions.
SELECT *
FROM supplier S, purchase_order PO
WHERE S.supplier_no = PO.supplier_no
This SQL statement selects only those tuples from the relations, Supplier and Purchase_Order, where the supplier_no in the Supplier relation matches the supplier_no in the Purchase_Order relation, and creates a new relation containing tuples consising of attributes from both relations.

15. EXAMPLE JOIN
For example, if the original relations, Supplier and Purchase_Order, contained the following data:
Supplier
supplier_no
name
address
contact_name
tel_no 1
Phones 4 Everyone Ltd
Unit 10 Count Drive Northampton NH10 4EX
Mr Smith 2
MP Tradesales
3 Clarence Drive Wolverhampton WX10 2GT
Mrs. Applegate 3
Buzz Mobiles Ltd
10 Green Gate Norwich NA2 6DT
Peter Taverham 4
Phone Accessories Direct
Warwick Crescent Leicester LE10 3AT
Mr Studley 5
Orion Express
2 Lindridge Industrial Estate Lindridge LD12 4TD
Mr Wheathill 6
Activity Phones Ltd.
Unit 8 Knowle Industrial Estate Knowle KN22 3AX
Mr Solihull

16. EXAMPLE JOIN And Purchase purchase_order_no supplier_no date 1 2 19 24 6

17. EXAMPLE JOIN
Then the result of the join would be a new relation with the structure shown below and containing the tuples listed below:
Supplier
_no
name
address
contact_name
tel_no
Purchase
_order_ no
date 1
Phones 4
Everyone Ltd
Unit 10 Count Drive Northampton NH10 4EX
Mr Smith
01637
890226 2
MP Tradesales
3 Clarence Drive Wolverhampton WX10 2GT
Mrs. Applegate
01456
789267 19 6
Activity Phones Ltd.
Unit 8 Knowle Industrial Estate Knowle KN22 3AX
Mr Solihull
01237
228999 24

18. EXAMPLE JOIN
This is an example of an equijoin, where the condition for matching tuples from both relations depends on the values of the matching field, or fields, in both relations being equal.
A natural join is a special case of an equijoin, where all attributes that have the same name in the relations to be joined form the basis of the join.

19. EXAMPLE JOIN The equivalent relational algebra expression follows:
Supplier S.supplier_no=PO.supplier_no Purchase_Order
This shows the name of the first relation, Supplier, followed by JOIN operator, followed by the condition controlling the join, which specifies which attributes, fields, or columns are to be matched, followed by the name of the second relation, Purchase_Order, to which the join will be applied.

20. EXAMPLE CROSS PRODUCT
This SQL statement combines the tuples from two relations to create new tuples containing attributes from both original relations. Every tuple from one relation is combined with every tuple from the other relation.
SELECT *
FROM supplier S, purchase_order PO
What happens if we apply this given the original data?
The equivalent relational algebra expression follows:
Supplier × Purchase_Order
This shows the name of the first relation, followed by the CROSS PRODUCT operator, followed by the name of the second relation, to which the cross product will be applied.

21. CONDITION JOIN
Contains those tuples of the cross product that satisfy some condition
The result schema is the same as that of the cross product
However there are fewer tuples than cross product ~ might be computationally more efficient
E.g. S S1.supplier_no < P.supplier_no P

22. EQUI-JOIN
A special case of condition join where the condition cond contains only equalities
E.g. S S1.supplier_no = P.supplier_no P
Result schema similar to cross product, but only one copy of fields for which equality is specified

23. Work through Examples
Taking into account the sailors database previously
used in class then write the possible relational
algebra statements for:
Find the names of the sailors who’ve reserved boat #103
Find the names of the sailors who’ve reserved a red boat
Find all sailors who have reserved a red or green boat