Relational Model and Relational Algebra - Lecture 3 - Introduction to Databases (1007156ANR)

2 December 2005
Introduction to Databases
Relational Model and Relational Algebra
Prof. Beat Signer
Department of Computer Science
Vrije Universiteit Brussel
http://www.beatsigner.com

Beat Signer - Department of Computer Science - bsigner@vub.ac.be 2March 3, 2017
Relational Model
 Theory for data management developed
by Edgar F. Codd while working at IBM
 Edgar F. Codd, A Relational Model of Data for
Large Shared Data Banks, Communications
of the ACM 13(6), June 1970
 data independence
- between logical and physical level
 set-based query language
- relational query language
 normalisation
- avoid redundancy
 IBM first did not implement the relational model in order
to "protect" their IMS/DB revenues
Edgar F. Codd

Relational Model …
 IBM's System R (1974) was a DBMS prototype
implementing Codd's relational model
 first implementation of the Structured English Query
Language (SEQUEL)
- later renamed to Structured Query Language (SQL)
 The System R prototype finally led to the development of
different commercial DBMSs including IBM's DB2,
Oracle or Microsoft SQL Server

Relational Database
 A relational database consists of a number of tables
 each table row defines a relationship between a set of values
 There is an analogy between the concept of a table
(collection of relationships) and the mathematical
concept of a relation
 in the following we therefore talk about relations instead of tables
  a relational database consists of a collection of relations
customerID name street postcode city
1 Max Frisch Bahnhofstrasse 7 8001 Zurich
2 Eddy Merckx Pleinlaan 25 1050 Brussels
... ... ... ... ...

Relational Database ...
 Information is normally partitioned into different relations
since the storage in a single relation would lead to
 a replication of information (redundancy)
 a large number of necessary null values
 While tables are used at the logical level, different
storage structures can be used at the physical level
 data independence

Relation
 The column headers of a table are called attributes and
for each attribute ai there is a set of permitted values
called the domain Di of ai
 Given the domains D1, D2,..., Dn, a relation r is defined as
a subset of the cartesian product D1D2...Dn
name street ... city
Max Frisch Bahnhofstrasse 7 ... Zurich
... ... ... ...
D1 D2 Dn
t1
tm
a1 a2 an
attributes
tuples
relation r
degree
cardinality

Relation ...
 A relation r is a set of n-ary tuples ti = (a1, a2,..., an) where
each aiDi
 the number of attributes is called a relation's degree
 the number of tuples is called a relation's cardinality
 Since a relation is a set of tuples, the order of the tuples
is irrelevant
 each tuple is distinctive (no duplicate tuples)
 The order of attributes is irrelevant
 The domain of each attribute has to be atomic
 all members of the domain have to be indivisible units
 a relation with only atomic values is normalised (first normal form)

Relation ...
 Multiple attributes can have the same domain Di
 we will see later how the domain types can be defined (in SQL)
 A tuple variable is a variable that stands for a tuple
 its domain is defined by the set of all tuples
 The special null value is part of any domain Di
 used to represent an unknown or non-existing value
 null values are not easy to handle (not the same as empty string)
- avoid null values whenever possible

Example of a Relation
 Let us assume that we have the following attributes with
their domains
 name = {Merckx, Frisch, Botta, ...}
 street = {Bahnhofstrasse, Pleinlaan, Via Nassa, ...}
 city = {Zurich, Brussels, Lugano, Paris, ...}
 Then
 r = {(Merckx, Pleinlaan, Brussels),
(Frisch, Bahnhofstrasse, Zurich),
(Botta, Via Nassa, Lugano),
(Botta, Bahnhofstrasse, Lugano)}
forms a relation over namestreetcity

Relational Database Example
1 Max Frisch Bahnhofstrasse 7 8001 Zurich
2 Eddy Merckx Pleinlaan 25 1050 Brussels
5 Claude Debussy 12 Rue Louise 75008 Paris
53 Albert Einstein Bergstrasse 18 8037 Zurich
8 Max Frisch ETH Zentrum 8092 Zurich
cdID name duration price year
1 Falling into Place 2007 17.90 2007
2 Carcassonne 3156 15.50 1993
customer
cd

Relational Database Example ...
orderID customerID cdID date amount status
1 53 2 13.02.2010 2 open
2 2 1 15.02.2010 1 delivered
order
supplierID name city
5 Max Frisch Zurich
2 Mario Botta Lugano
supplier
Customer (customerID, name, street, postcode, city)
CD (cdID, name, duration, price, year)
Order (orderId, customerID, cdID, date, amount, status)
Supplier (supplierID, name, city)
relational database schema

Database Schema
 The logical design of a database is defined by the
database schema
 A database instance is a snapshot of the data stored in a
database at a given time
 A relation schema R = (A1, A2,..., An) is defined by a list of
attributes
 by convention, the name of a relation schema starts with an
uppercase letter (in contrast to relations)
- e.g. Customer (customerID, name, street, postcode, city)
 a relation can be defined based on the relation schema
- e.g. customer = (Customer)
 a relation instance contains the relation's actual values (tuples)

Keys
 For the keys we use the same terminology as
introduced earlier for the ER model
 K  R is a superkey of R if the values of K uniquely
identify any tuple of possible relation instances r(R)
 t1 and t2  r and t1  t2  t1[K]  t2[K]
 e.g. {cdID} and {cdID, name} are both superkeys of CD
 K is a candidate key if K is minimal
 e.g. {cdID} is a candidate key of CD
 The DB designer has to choose one of the candidate
keys for each relation schema as primary key
 if possible, the value of a primary key should not change or only in
very rare cases

Foreign Keys
 A relation schema may have one or multiple attributes
that correspond to the primary key of another relation
schema and are called foreign keys
 e.g. customerID is a foreign key of the order relation schema
 note that a foreign key of the referencing relation can only contain
values that occur in the primary key of the referenced relation or it
must be null  referential integrity
customerID
name
street
postcode
city
cdID
name
duration
price
year
orderID
customerID
cdID
date
amount
status
customer cd
order
schema diagram

 A query language is a language that is used to access
(read) information stored in a database as well as to
create, update and delete information (CRUD)
 There are different types of query languages
 procedural
- e.g. relational algebra
 declarative
- e.g. Structured Query Language (SQL)
 The relational algebra as well as the tuple and domain
relational calculus form the basis for "higher level" query
languages (e.g. SQL)
Query Language

Relational Algebra
 The relational algebra consists of six fundamental
operations
 unary operations
 selection: s
 projection: p
 rename: r
 binary operations
 union: 
 set difference: -
 cartesian product: 
 An operator takes one (unary) or two (binary) relations
as input and returns a new single relation as output

Relational Algebra ...
 Based on the six fundamental operations, we can define
additional relational operations (add no additional power)
 set intersection: 
 natural join: ⋈
 theta join: q
 equijoin: q
 semijoin: ⋉
 division: 
 assignment: 
 Extended operations with additional expressiveness
 generalised projection
 aggregate function
 outer join

Selection (s)
 Goal: Select specific tuples (rows) from a relation (table)
 sp(r) = {t | t  r and p(t)}
 p is a selection predicate
- consists of terms pi connected by an and (), or () or not ()
 a term pi has the form attributem operator attributen or constant
- the available operators are: =, , >, <, , 
 Examples
 "Find all tuples in the customer relation that are in the city of
Zurich and have a postcode greater than 8010."
 scity = "Zurich"  postcode > 8010 (customer)
8 Max Frisch ETH Zentrum 8092 Zurich

Selection (s) ...
 Examples ...
 A selection predicate may contain a comparison between two
attributes
 "Find all tuples in the cd relation with a value for the duration
attribute that is equal to the year of release."
 sduration = year (cd)
 Attention: The selection operation in relational algebra
has a different meaning than the SELECT statement used
in SQL
 SELECT in SQL corresponds to a projection in relational algebra

Projection (p)
 Goal: Select specific attributes (columns) from a relation
 pA1, A2,..., Am
(r)
 returns a relation instance that only contains the columns for
which an attribute Ai has been listed
 duplicate tuples are removed from the resulting relation (set)
 Example
 "Return the name and city of all tuples in the customer relation."
 pname,city (customer)
name city
Max Frisch Zurich
Eddy Merckx Brussels
Claude Debussy Paris
Albert Einstein Zurich

Composition of Relational Operations
 Since the result of a relational operation is a new single
relation instance, multiple operations can be combined
 Example
 "Find the names of all customers who live in Zurich."
 pname (scity = "Zurich" (customer))
name
Max Frisch
Albert Einstein

Union ()
 Goal: Unify tuples from two relations
 r  s = {t | t  r or t  s}
 r and s must have the same degree (same number of attributes)
 the corresponding attribute domains must have a compatible type
 Example
 "Find the names of persons who are either customers or suppliers."
 pname (customer)  pname (supplier)
name
Max Frisch
Eddy Merckx
Claude Debussy
Albert Einstein
Mario Botta

Set Difference (-)
 Goal: Find tuples that are in one relation but not in
another relation
 r - s = {t | t  r and t  s}
 r and s must have the same degree
 finds tuples that are in a relation r but not in another relation s
 Example
 "Find the names of suppliers who are no customers."
 pname (supplier) - pname (customer)
name
Mario Botta

Cartesian Product ()
 Goal: Combine information from any two relations
 r  s = {tu | t  r and u  s}
 the attribute names of r(R) and s(S) have to be distinct
- attributes with the same name have to be renamed via the rename operator r
 Example
 pname (customer)  pcity (customer)
name city
Max Frisch Zurich
Max Frisch Brussels
Max Frisch Paris
Eddy Merckx Zurich
Eddy Merckx Brussels
... ...

Cartesian Product () ...
 Example ...
 "List the names of all customers with at least one order."
 pname( scustomer.customerID = order.customerID (customer  order) )
 Note that we will later see another operator (natural join)
for the combination of two tables based on common
attributes
name
Eddy Merckx
Albert Einstein

Rename (r)
 Goal: Rename a relation or an attribute
 rx(E)
 renames the result of expression E to x
 rx(A1,A2,..., An)
(E)
 renames the result of expression E to x and renames the
attributes to A1,A2,...,An
 Example
 rperson(name,location) (pname,city (customer))
name location
Max Frisch Zurich
... ...
person

Rename (r) ...
 Example
 "Find the price of the most expensive CD in the cd relation."
 Pprice (cd) - Pcd.price(scd.price < d.price (cd  rd (cd)))
 We will later see another operator (aggregate function)
that can be used to simplify this type of query for a max
value
price
17.90

Formal Definition
 A basic relational algebra expression is either a relation
from the database or a constant relation
 The set of all relational algebra expressions is defined by
 E1  E2
 E1 - E2
 E1  E2
 sp(E1)
 pA1,A2,..., Am
(E1)
 rx(A1,A2,..., An)
(E1)

Set Intersection
 r  s = {t | t  r and t  s}
 r and s must have the same degree
 Note that the set intersection can be implemented via a
pair of set difference operations
 r  s = r - (r - s)
 Example
 "Find the names of people who are customers and suppliers."
 pname (customer)  pname (supplier)
name
Max Frisch

Natural Join (⋈)
 r ⋈ s = pR  S (sr.A1= s.A1  r.A2= s.A2 ... r.An= s.An
(r  s))
 where R  S = {A1, A2,..., An}
 Keep all tuples of the cartesian product r  s that have the
same value for the shared attributes of r(R) and s(S)
 the natural join is an associative operation: (r ⋈ s) ⋈ t = r ⋈ (s ⋈ t)
 Example
 "List the name and street of customers whose order is still open."
 pname, street(sstatus="open"(order ⋈ customer))
 note that we could do the selection first (more efficient)
- we will review this later when discussing query optimisation
name street
Albert Einstein Bergstrasse 18

Theta Join (q) and Equijoin
 r ⋈p s = sp (r  s)
 where the predicate p is of the form r.Ai q s.Aj and q is a
comparison operator (=, , >, <, , )
 Example
 customer ⋈postcode  duration (rcd(cdID,cdName,duration,price,year) (cd))
 An equijoin (r ⋈p s) is a special form of a theta join where
only the equality operator (=) may be used
 note that a natural join is a equijoin over all common attributes of
a relation r and s
customerID name ... cdID cdName ...
2 Eddy Merckx ... 1 Falling into Place ...
2 Eddy Merckx ... 2 Carcassonne ...

Semijoin (⋉)
 r ⋉ s = pA1, A2,..., An
(r ⋈ s)
 where A1, A2,..., An are all the attributes of r
 Example
 "List the customers whose order is still open."
 customer ⋉ sstatus="open" (order)
customerID Name Street Postcode city

Division ()
 r  s = {t | t  pR-S(r) and " u  s  tu  r}
 for the relations r(R) and s(S) and S  R
 suited for queries that include the phrase "for all"
 Example
 r  s
A B C D
a 3 a 1
a 1 a 3
a 3 b 3
b 1 a 1
c 4 b 3
b 2 b 2
c 4 a 1
r
C D
b 3
a 1
A B
a 3
c 4
s
r  s

Assigment
 variable  E
 Works like an assigment in programming languages
 Assigments must always be made to temporaray relation
variables
 no database modification
 Example
 temp1  pname, street(sstatus="open"(order ⋈ customer))
 temp2  pname,street (scity = "Brussels" (customer))
 result  temp1  temp2
name street
Eddy Merckx Pleinlaan 25
Albert Einstein Bergstrasse 18

Generalised Projection (p)
 pF1, F2,..., Fm
(E)
 generalisation of the project operation that supports the projection
to arithmetic expressions F1, F2,..., Fm
 Example
 "Show a list of the names of all CDs together with a reduced
price (80% of the original price)."
 pname, price  0.8 as reducedPrice (cd)
name reducedPrice
Falling into Place 14.32
Carcassonne 12.72

Aggregate Function
 G1, G2,..., Gm
G F1(A1), F2(A2),..., Fn(An) (E)
 takes a collection of values as input and returns a single value
based on the following operations
- min: minimum value
- max: maximum value
- sum: sum of values
- count: number of value
- avg: average value
 G1, G2,..., Gm lists the attributes on which to group
- can be empty
 Fi are the aggregate functions
 by default, duplicates are not elimiated before aggregating
- can use the keyword distict to eliminate duplicates (e.g. sum-distinct)

Aggregate Function ...
 Examples
 "List the average amount of items in an order."
 G avg(amount)(order)
 "List the number of customers for each city."
 cityG count(customerID) as customerNo (customer)
avg(amount)
1.5
city customerNo
Zurich 3
Brussels 1
Paris 1

Outer Joins
 The left outer join (⟕), right outer join (⟖) and full outer
join (⟗) are extensions of the natural join operation
 Computes the natural join and then adds the tuples from
one relation that do not match the other relation
 filled up with null values
 Example
 customer ⟕ order
customerID name ... orderID cdID ...
53 Albert Einstein ... 1 2 ...
2 Eddy Merckx ... 2 1 ...
5 Claude Debussy ... null null ...
... ... ... ... ... ...

Outer Joins ...
 left outer join (⟕)
 keeps all tuples of the left relation
 non-matching parts filled up with null values
 right outer join (⟖)
 keeps all tuples of the right relation
 full outer join (⟗)
 keeps all tuples of both relations

Null Values
 A null value means that the value is unknown or
nonexistent
 Primary key attributes can never be null (entity integrity)
 The result of an arithmetic operation that involves a null
value is null
 For grouping and duplicate elimination null values are
treated like other values (two null values are the same)
 Null values are ignored in aggregate functions

Database Modifications
 A database may be modified by using one of the
following operations
 insert
 update
 delete
 These operations are defined via the assigment operator

Insert
 r  r  E
 To insert new data into a relation we can
 explicitly specify the tuples to be inserted
 write a query and insert the result tuple set
 Example
 cd  cd  {(3, "Chromatic", 3012, 16.50, 1994)}
2 Carcassonne 3156 15.50 1993
3 Chromatic 3012 16.50 1994

Update
 r  pF1, F2,..., Fn
(r)
 Update specific values in a tuple r
 Fi is either the old value of r or a new value if the
attribute has to be updated
 Example
 "Increase the price of all CDs by 10%."
 cd  pcdID, name, duration, price  1.1, year (cd)
2 Carcassonne 3156 17.05 1993
3 Chromatic 3012 18.15 1994

Delete
 r  r - E
 A delete is expressed similar to a query except that the
result is removed from the database
 Cannot remove single attributes
 Example
 "Remove the CD with the name 'Carcassonne' from the
database."
 cd  cd - sname = "Carcassonne" (cd)
3 Chromatic 3012 18.15 1994

Homework
 Study the following chapters of the
Database System Concepts book
 chapter 2
- Relational Model
 chapter 6
- section 6.1
- Relational Algebra

Exercise 3
 Relational model
 Relational algebra

References
 A. Silberschatz, H. Korth and S. Sudarshan,
Database System Concepts (Sixth Edition),
McGraw-Hill, 2010
 E. F. Codd, A Relational Model of Data for Large Shared
Data Banks, Communications of the ACM 13(6),
June 1970

2 December 2005
Next Week
Relational Database Design

Relational Model and Relational Algebra - Lecture 3 - Introduction to Databases (1007156ANR)

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