1. Relational Model
B.Ramamurthy
5/28/2019
B.Ramamurthy

2. Introduction
Relational database theory assist in definition of databases and in efficient processing of databases and in efficient processing of user requests for information from a database.
ER model defined the problem and leads us to the tables. Now we need to define, manipulate and use the tables.. Relational model provides the foundation for this.
5/28/2019
B.Ramamurthy

3. Topics for Discussion Structure of Relational database
Relational Algebra
Tuple Relational Calculus
Domain Relational Calculus
Extended Relational-Algebra-Operations
Modifications of the Database
Views
Summary
5/28/2019
B.Ramamurthy

4. Relational Database
A relational database consists of a collection of tables, each of which is assigned a unique name.
A table will be referred to as a relation in Relational DB.
Tuple is the term for row in a table.
On to more theoretical stuff …
5/28/2019
B.Ramamurthy

5. Basic Structure Given sets A1, A2, A3…An, a relation r is a subset of
A1 X A2 X A3 X ….X An
A relation is a set of n-tuples (a1,a2,a3,..an) where ai Є Ai.
Example : if
customer-name = {Jones, Smith, Curry, Lindsay}
customer-street = {Main, North, Park}
customer-city = {Harrison, Rye, Pittsfield}
Then r = {(Jones,Main,Harrison), (Smith,North,Rye), {Curry,North,Rye), (Lindsay, Park, Pittsfield)} is a relation over
customer-name X customer-street X customer-city
5/28/2019
B.Ramamurthy

6. Relation Schema
(This concept is analogous to class, instance of a class in OOP)
A1, A2, A3.. An are attributes
R = (A1, A2, A3, …An) is a relation schema.
Example : Customer-schema = (customer-name, customer-street, customer-city)
r(R) is a relation on the relational schema R
Example : customer(Customer-schema)
5/28/2019
B.Ramamurthy

7. More Examples Branch-scheme= (banch-name, branch-city,assets)
Relation is a table of specific branches
branch-name branch-city assets
Downtown Brooklyn
Redwood Palo Alto
North Town Rye
5/28/2019
B.Ramamurthy

8. Relation Instance
Current values (relational instance) of a relation are specified by a table.
An element t of r is a tuple: represented by a row in the “relation” table
See the above example
5/28/2019
B.Ramamurthy

9. Query Language
Language in which user requests information from a database.
Categories of languages: procedural, non procedural
Pure Languages:
Relational algebra
Tuple relational calculus
Domain relational calculus
Pure languages form the basis of query languages.
5/28/2019
B.Ramamurthy

10. Relational Algebra Procedural Language Six fundamental operations
Select
Project
Union
Set difference
Cartesian product
Rename
The operators take two or more relations as inputs and give a new relation as a result.
5/28/2019
B.Ramamurthy

11. Select Operation
Select operation selects tuples that satisfy a given predicate.
Lowercase sigma is used to denote selection.
The predicate appears as subscript to p
The argument relation is given in parenthesis.
Therefor notation for Select operation is p (r)
5/28/2019
B.Ramamurthy

12. Select: Formal Definition
P ( r ) = {t | t Є r and P(t)}
where t is a tuple in r that satisfies P(t)
P(t) is a formula in propositional calculus, dealing with terms of the form:
<attribute> op <attribute> or constant
op represents an operator in { =, <>, > , >=, <=}
simple terms can be connected by and, or, not
5/28/2019
B.Ramamurthy

13. Select: Example c Relation r : A B C D   1 7 5 12 3 23 10  
A= B and D > 5 (r )   23 10
5/28/2019
B.Ramamurthy

14. Project Operation
Suppose you are interested in retrieving only a few attributes of a relation.
A “Project” operation can be used.
It is a unary operation.
Basically a vertical subset of a table is returned.
Duplicate rows are removed from the result.
5/28/2019
B.Ramamurthy

15. Project: Formal Definition
A1,A2,…Ak ( r )
where A1, A2, …Ak are attribute names and r is a relation name.
Result has k columns, each column representing an attribute.
5/28/2019
B.Ramamurthy

16. Project: Example A B C D Relation r :   1 7 5 7 12 3   23 10  
A,B ( r )
5/28/2019
B.Ramamurthy

17. Union Operation Binary operation. Notation : r U s
Definition: r U s = {t | t Є r or t Є s}
Constraints are:
1. r and s must have same arity (same number of attributes)
2. The attribute domains must be compatible (ex: 2 nd column of r is of String type and so should 2nd column of s)
5/28/2019
B.Ramamurthy

18. Union: Example  1  2 s r   2 3 r U s  1   2 1  3 A B A B A B
5/28/2019
B.Ramamurthy

19. Other Important Operations
We discussed Join, Division, Cartesian Product, Rename and Assignment operations.
For each of these we studied:
1. The symbol
2. Formal expression
3. An example in tabular form
4. A sample query that needs (or uses) this operation.
5/28/2019
B.Ramamurthy

20. Tuple Relational Calculus
A nonprocedural query language, where each query is of the form
{ t | P(t) }
It is the set of all tuples t such that predicate P is true for t.
t is a tuple variable; t[ A] denotes the value of tuple t on attribute A and t  r denotes that tuple t is in relation r.
P is a formula similar to that of the predicate calculus.
5/28/2019
B.Ramamurthy

21. Sample Queries {t | t  loan  t[amount] > 1200}
Write the following in tuple relational calculus as well as in relational algebra.
Loan numbers such that loan-amount > 1200
Names of all customers who have a loan from the PerryRidge branch.
5/28/2019
B.Ramamurthy

22. Extended Relational Algebra Operations
Generalized projection: Apply arithmetic (or any) function to the regular projection.
Example : Credit info of customers.
Outer Join: Extension of join that deals with missing information. Lossless join.
We have left outer join and right outer join.
Aggregate functions: to take a collection of values and return a single value. Example : sum of a column.
5/28/2019
B.Ramamurthy

23. Modification of the database
Deletion, insertion, updating : all these major operations are actual a collection of lower level operations we have discussed.
Views: View definitions will also have only part of the data stored in the table depending on the request. Once again we apply the project , join and other operations we studied.
5/28/2019
B.Ramamurthy

24. Summary A relational database is made up of tables.
Each table is a relation.
Each row of a table is a relationship.
E-R diagram --> relation-schema--> tables --> RDB
Relational operations are used to extract data from the tables.
5/28/2019
B.Ramamurthy