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

http://www.beatsigner.com/

Beat Signer - Department of Computer Science - [email protected]

2 February 28, 2014

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


3 February 28, 2014

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


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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

... ... ... ... ...


5 February 28, 2014

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


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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 D1D2...Dn

name street ... city

Max Frisch Bahnhofstrasse 7 ... Zurich

... ... ... ...

D1 D2 Dn

t1

tm

a1 a2 an attributes

tuples

relation r

degree

cardinality


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Relation ...

A relation r is a set of n-ary tuples ti = (a1, a2,..., an) where

each aiDi 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)


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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


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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 namestreetcity


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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,..., GmG 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

Education

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