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Chapter 3: Relational Model and
Relational Algebra & Calculus
( [S] Chp. 2 and 5 )
 Structure of Relational Databases
 Relational Algebra
 Tuple Relational Calculus
 Domain Relational Calculus
 Extended Relational-Algebra-Operations
 Modification of the Database
 Views
Database System Concepts
3.1
©Silberschatz, Korth and Sudarshan
Why Study the Relational Model?
 Most widely used model.
• Vendors: IBM, Informix, Microsoft, Oracle, Sybase, etc.
 “Legacy systems” in older models
• E.G., IBM’s IMS
 Recent competitor: object-oriented model
• ObjectStore, Versant, Ontos
• A synthesis emerging: object-relational model
 Informix Universal Server, UniSQL, O2, Oracle, DB2
Database System Concepts
3.2
©Silberschatz, Korth and Sudarshan
Basic Structure
 Formally, given sets D1, D2, …. Dn, a relation r is a subset of
D1 x D2 x … x Dn
Each Di is a Domain
Thus a relation is a set of n-tuples (a1, a2, …, an) where each ai  Di
 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
Database System Concepts
3.3
©Silberschatz, Korth and Sudarshan
Attribute Types
 Each attribute of a relation has a name
 The set of allowed values for each attribute is called the domain of the
attribute
 Attribute values are (normally) required to be atomic, that is, indivisible
• E.g. multivalued attribute values are not atomic
• E.g. composite attribute values are not atomic
 The special value null is a member of every domain
 The null value causes complications in the definition of many operations
• we shall ignore the effect of null values in our main presentation and consider
their effect later
Database System Concepts
3.4
©Silberschatz, Korth and Sudarshan
Relation Schema
 A1, A2, …, An are attributes
 R = (A1, A2, …, An ) is a relation schema
E.g. Customer-schema =
(customer-name, customer-street, customer-city)
 r(R) is a relation on the relation schema R
E.g.
customer (Customer-schema)
Our common notation: R – relational schema, r- relation instance
When we say “relation” we will mean relation instance or relation
schema depending on the context
Database System Concepts
3.5
©Silberschatz, Korth and Sudarshan
Relation Instance
 The current values (relation instance) of a relation are
specified by a table
 An element t of r is a tuple, represented by a row in a table
 The number of attributes is the relation degree, the number of
rows is the relation cardinality
attributes
(or columns)
customer-name customer-street
Jones
Smith
Curry
Lindsay
Main
North
North
Park
customer-city
Harrison
Rye
Rye
Pittsfield
tuples
(or rows)
customer
Database System Concepts
3.6
©Silberschatz, Korth and Sudarshan
Relations are Unordered
 Order of tuples is irrelevant (tuples may be stored in an arbitrary order)
 E.g. account relation with unordered tuples
Database System Concepts
3.7
©Silberschatz, Korth and Sudarshan
Example Instance of Students Relation
sid
53666
53688
53650
name
login
Jones jones@cs
Smith smith@eecs
Smith smith@math
age
18
18
19
gpa
3.4
3.2
3.8
Cardinality = 3, degree = 5, all rows distinct
Database System Concepts
3.8
©Silberschatz, Korth and Sudarshan
Database
 A database consists of multiple relations
 Information about an enterprise is broken up into parts (usually
an entity), with each relation storing one part of the information
E.g.: account : stores information about accounts
depositor : stores information about which customer
owns which account
customer : stores information about customers
 Storing all information as a single relation such as
bank(account-number, balance, customer-name, ..)
results in
• repetition of information (e.g. two customers own an account)
• the need for null values (e.g. represent a customer without an
account)
 Normalization theory (Chapter 7) deals with how to design
relational schemas
Database System Concepts
3.9
©Silberschatz, Korth and Sudarshan
E-R Diagram for the Banking Enterprise
Database System Concepts
3.10
©Silberschatz, Korth and Sudarshan
Relation Schema of the Bank Example
Database System Concepts
3.11
©Silberschatz, Korth and Sudarshan
The customer Relation
Database System Concepts
3.12
©Silberschatz, Korth and Sudarshan
The depositor Relation
Database System Concepts
3.13
©Silberschatz, Korth and Sudarshan
The Account Relation
Database System Concepts
3.14
©Silberschatz, Korth and Sudarshan
Keys
 Let K  R
 K is a superkey of R if values for K are sufficient to identify a unique tuple
of each possible relation r(R)
• by “possible r” we mean a relation r that could exist in the enterprise we are
modeling.
• Example: {customer-name, customer-street} and
{customer-name}
are both superkeys of Customer, if no two customers can possibly have the
same name.
 K is a candidate key if K is minimal
• Example: {customer-name} is a candidate key for Customer.
• Since it is a superkey, and no subset of it is a superkey.
 Assuming no two customers can possibly have the same name.
Database System Concepts
3.15
©Silberschatz, Korth and Sudarshan
Determining Keys from E-R Sets
 Strong entity set.
• The primary key of the entity set becomes the primary key of the relation.
 Weak entity set.
• The primary key of the relation consists of the union of the primary key of the
strong entity set and the discriminator of the weak entity set.
 Relationship set.
• The union of the primary keys of the related entity sets becomes a super key of
the relation.
 For binary many-to-one relationship sets
 The primary key of the “many” entity set becomes the relation’s primary
key.
 For one-to-one relationship sets, the relation’s primary key can be that of
either entity set.
 For many-to-many relationship sets
 The union of the primary keys becomes the relation’s primary key
Database System Concepts
3.16
©Silberschatz, Korth and Sudarshan
‫המודל הטבלאי‬
‫‪ ‬פותח לראשונה על ידי ‪ Codd‬בשנת ‪.1970‬‬
‫‪ ‬הפך להיות המודל השליט בשנות ה‪ 08 -‬וה‪.90 -‬‬
‫‪ ‬רוב המערכות המסחריות מבוססות עליו‪:‬‬
‫• ‪DB2, ORACLE, SYBASE, Informix‬‬
‫• מדוע?‬
‫‪ ‬פשוט‪ ,‬אלגנטי ומבוסס תאורטית‪.‬‬
‫‪©Silberschatz, Korth and Sudarshan‬‬
‫‪3.17‬‬
‫‪Database System Concepts‬‬
‫ מושגים‬- ‫המודל הטבלאי‬
.‫ – קבוצה של ערכים אטומיים‬Domain – ‫ תחום‬
(Intention) R(A1, A2…An) (scheme) Relation – ‫ טבלה‬
.‫ עם ערכים הלקוחים מתחום מסויים‬Attribute – ‫ זו תכונה‬Ai

(Extension) r(t1, t2…tm) (Instance) Relation – ‫ טבלה‬
tuple – ‫ שורה‬
t=(v1, v2…vn)
.t ‫ בשורה‬i ‫ – ערך של תכונה‬Vi

.Degree or Arity – ‫ מספר העמודות בטבלה – דרגה‬
Cardinality - ‫ השורות‬.‫ מס‬
Database System Concepts
3.18
©Silberschatz, Korth and Sudarshan
‫המודל הטבלאי ‪ -‬תכונות‬
‫‪ ‬אין חשיבות לסדר השורות‪.‬‬
‫‪ ‬אין חשיבות לסדר העמודות‪.‬‬
‫‪ ‬אין שתי שורות זהות בטבלה‪.‬‬
‫‪ ‬קבוצת תכונות המזהה שורה באופן יחיד – סופר מפתח – ‪.Super-Key‬‬
‫•‬
‫קבוצה מינימלית כזו נקראת ‪.Candidate key‬‬
‫•‬
‫אחת הקבוצות הללו נקראת ה – ‪.Primary key‬‬
‫‪©Silberschatz, Korth and Sudarshan‬‬
‫‪3.19‬‬
‫‪Database System Concepts‬‬
‫המודל הטבלאי – סימונים‬
‫‪ ‬טבלאות (סכימה)‬
‫– ‪Q, R, S‬‬
‫‪ ‬טבלאות (מופע )‬
‫– ‪q, r, s‬‬
‫‪ ‬שורות‬
‫– ‪t, u, v‬‬
‫•‬
‫]‪ – t[Ai‬הערך של תכונה – ‪ Ai‬עבור ‪.t‬‬
‫•‬
‫לדוגמה‪ – Ai :‬עבור שורה ‪.t(SSN, GPA) t‬‬
‫‪ ‬במידה ושם התכונה לא יחיד – ‪R. Ai‬‬
‫•‬
‫לדוגמה‪STUDENT.NAME :‬‬
‫‪ ‬ערך לא קיים ‪Null -‬‬
‫‪©Silberschatz, Korth and Sudarshan‬‬
‫‪3.20‬‬
‫‪Database System Concepts‬‬
‫המודל הטבלאי ‪ -‬אילוצים‬
‫‪ ‬תחומים‪:‬‬
‫כל הערכים בתחום הם אטומיים (או ‪.)Null‬‬
‫‪ ‬מפתחות‪:‬‬
‫לכל טבלה יש לפחות מפתח אחד‪ ,‬לאחד‬
‫מהם קוראים מפתח ראשי – ‪.primary‬‬
‫• ערך של תכונה במפתח לא יכול להיות ‪.Null‬‬
‫‪ ‬מפתח זר‪:‬‬
‫לתכונה של מפתח זר בטבלה יש אותם תחומים כמו‬
‫לתכונות של מפתח ראשי בטבלה אחרת‪.‬‬
‫‪ ‬אלוץ מפתח זר‪ FK constraint – referential integrity constraint :‬ערך‬
‫של מפתח זר הוא או ‪ Null‬או שקיים כערך של מפתח ראשי‬
‫בטבלה המתאימה‪.‬‬
‫• משמעות‪:‬‬
‫קשר‬
‫• חשיבות‪:‬‬
‫הוספה‪ ,‬עדכון ובטול‬
‫• נראה הגדרה פורמלית מייד‬
‫‪©Silberschatz, Korth and Sudarshan‬‬
‫‪3.21‬‬
‫‪Database System Concepts‬‬
‫המודל הטבלאי – אילוצים נוספים‬
‫‪.1‬‬
‫אילוצי ערך‪:‬‬
‫‪0 < salary < 100,000‬‬
‫‪.2‬‬
‫אילוצים בין שדות‪:‬‬
‫‪QTY_SUPPLIED ≤ QTY_ORDERED‬‬
‫‪.3‬‬
‫תלויות פונקציונליות‪:‬‬
‫‪NAME -> AGE‬‬
‫אם אותו שם מופיע פעמיים אז הוא מופיע עם אותו גיל‪.‬‬
‫‪.4‬‬
‫‪ – SQL Integrity Constraints‬אילוצים מורכבים‬
‫‪©Silberschatz, Korth and Sudarshan‬‬
‫‪3.22‬‬
‫‪Database System Concepts‬‬
‫בסיס נתונים טבלאי ‪ -‬הגדרה‬
‫‪ ‬אוסף הטבלאות‪ ,‬תכונות‪ ,‬תחומים‪ ,‬מפתחות ואילוצי שלמות וכל המופעים של‬
‫הטבלאות הללו המקיימים את ההגדרות והאילוצים הללו‪.‬‬
‫‪©Silberschatz, Korth and Sudarshan‬‬
‫‪3.23‬‬
‫‪Database System Concepts‬‬
Domain Constraints
 Integrity constraints guard against accidental damage to the
database, by ensuring that authorized changes to the database do
not result in a loss of data consistency.
 Domain constraints are the most elementary form of integrity
constraint.
 They test values inserted in the database, and test queries to
ensure that the comparisons make sense.
 New domains can be created from existing data types
• E.g. create domain Dollars numeric(12, 2)
create domain Pounds numeric(12,2)
 We cannot assign or compare a value of type Dollars to a value of
type Pounds.
• However, we can convert type as below
(cast r.A as Pounds)
(Should also multiply by the dollar-to-pound conversion-rate)
Database System Concepts
3.24
©Silberschatz, Korth and Sudarshan
Domain Constraints (Cont.)
 The check clause in SQL-92 permits domains to be restricted:
• Use check clause to ensure that an hourly-wage domain allows only
values greater than a specified value.
create domain hourly-wage numeric(5,2)
constraint value-test check(value > = 4.00)
• The domain has a constraint that ensures that the hourly-wage is
greater than 4.00
• The clause constraint value-test is optional; useful to indicate which
constraint an update violated.
 Can have complex conditions in domain check
• create domain AccountType char(10)
constraint account-type-test
check (value in (‘Checking’, ‘Saving’))
• check (branch-name in (select branch-name from branch))
Database System Concepts
3.25
©Silberschatz, Korth and Sudarshan
Referential Integrity
 Ensures that a value that appears in one relation for a given set of
attributes also appears for a certain set of attributes in another
relation.
• Example: If “Perryridge” is a branch name appearing in one of the
tuples in the account relation, then there exists a tuple in the branch
relation for branch “Perryridge”.
 Formal Definition
• Let r1(R1) and r2(R2) be relations with primary keys K1 and K2
respectively.
• The subset  of R2 is a foreign key referencing K1 in relation r1, if for
every t2 in r2 there must be a tuple t1 in r1 such that t1[K1] = t2[].
• Referential integrity constraint is also called subset dependency since
its can be written as
 (r2)  K1 (r1)
Database System Concepts
3.26
©Silberschatz, Korth and Sudarshan
Referential Integrity in the E-R Model
 Consider relationship set R between entity sets E1 and E2. The
relational schema for R includes the primary keys K1 of E1 and
K2 of E2.
Then K1 and K2 form foreign keys on the relational schemas for
E1 and E2 respectively.
E1
R
E2
 Weak entity sets are also a source of referential integrity
constraints.
• For the relation schema for a weak entity set must include the
primary key attributes of the entity set on which it depends
Database System Concepts
3.27
©Silberschatz, Korth and Sudarshan
Checking Referential Integrity on
Database Modification
 The following tests must be made in order to preserve the
following referential integrity constraint:
 (r2)  K (r1)
 Insert. If a tuple t2 is inserted into r2, the system must ensure
that there is a tuple t1 in r1 such that t1[K] = t2[]. That is
t2 []  K (r1)
 Delete. If a tuple, t1 is deleted from r1, the system must
compute the set of tuples in r2 that reference t1:
 = t1[K] (r2)
If this set is not empty
• either the delete command is rejected as an error, or
• the tuples that reference t1 must themselves be deleted
(cascading deletions are possible).
Database System Concepts
3.28
©Silberschatz, Korth and Sudarshan
Database Modification (Cont.)
 Update. There are two cases:
•
If a tuple t2 is updated in relation r2 and the update modifies values for
foreign key , then a test similar to the insert case is made:

Let t2’ denote the new value of tuple t2. The system must ensure
that
t2’[]  K(r1)
•
If a tuple t1 is updated in r1, and the update modifies values for the
primary key (K), then a test similar to the delete case is made:
1. The system must compute
 = t1[K] (r2)
using the old value of t1 (the value before the update is applied).
2. If this set is not empty
1. the update may be rejected as an error, or
2. the update may be cascaded to the tuples in the set, or
3. the tuples in the set may be deleted.
Database System Concepts
3.29
©Silberschatz, Korth and Sudarshan
Referential Integrity in SQL
 Primary and candidate keys and foreign keys can be specified as part of
the SQL create table statement:
• The primary key clause lists attributes that comprise the primary key.
• The unique key clause lists attributes that comprise a candidate key.
• The foreign key clause lists the attributes that comprise the foreign key and
the name of the relation referenced by the foreign key.
 By default, a foreign key references the primary key attributes of the
referenced table
foreign key (account-number) references account
 Short form for specifying a single column as foreign key
account-number char (10) references account
 Reference columns in the referenced table can be explicitly specified
• but must be declared as primary/candidate keys
foreign key (account-number) references account(account-number)
Database System Concepts
3.30
©Silberschatz, Korth and Sudarshan
Referential Integrity in SQL – Example
create table customer
(customer-name char(20),
customer-street char(30),
customer-city
char(30),
primary key (customer-name))
create table branch
(branch-name
char(15),
branch-city
char(30),
assets
integer,
primary key (branch-name))
Database System Concepts
3.31
©Silberschatz, Korth and Sudarshan
Referential Integrity in SQL – Example (Cont.)
create table account
(account-number
char(10),
branch-name char(15),
balance
integer,
primary key (account-number),
foreign key (branch-name) references branch)
create table depositor
(customer-name
char(20),
account-number
char(10),
primary key (customer-name, account-number),
foreign key (account-number) references account,
foreign key (customer-name) references customer)
Database System Concepts
3.32
©Silberschatz, Korth and Sudarshan
Cascading Actions in SQL
create table account
...
foreign key(branch-name) references branch
on delete cascade
on update cascade
...)
 Due to the on delete cascade clauses, if a delete of a tuple in
branch results in referential-integrity constraint violation, the
delete “cascades” to the account relation, deleting the tuple that
refers to the branch that was deleted.
 Cascading updates are similar.
Database System Concepts
3.33
©Silberschatz, Korth and Sudarshan
Cascading Actions in SQL (Cont.)
 If there is a chain of foreign-key dependencies across multiple
relations, with on delete cascade specified for each dependency,
a deletion or update at one end of the chain can propagate across
the entire chain.
 If a cascading update to delete causes a constraint violation that
cannot be handled by a further cascading operation, the system
aborts the transaction.
•
As a result, all the changes caused by the transaction and its
cascading actions are undone.
 Referential integrity is only checked at the end of a transaction
• Intermediate steps are allowed to violate referential integrity provided
later steps remove the violation
• Otherwise it would be impossible to create some database states, e.g.
insert two tuples whose foreign keys point to each other
 E.g. spouse attribute of relation
marriedperson(name, address, spouse)
Database System Concepts
3.34
©Silberschatz, Korth and Sudarshan
Referential Integrity in SQL (Cont.)
 Alternative to cascading:
• on delete set null
• on delete set default
 Null values in foreign key attributes complicate SQL referential
integrity semantics, and are best prevented using not null
• if any attribute of a foreign key is null, the tuple is defined to satisfy
the foreign key constraint!
Database System Concepts
3.35
©Silberschatz, Korth and Sudarshan
Foreign keys and Relationships
 In the relational model foreign keys represent relationships
 In contrast to primary (candidate) keys foreign keys may be null
representing partial relationship.
 One to one is represented as a foreign key in one of the
participating relations. Always prefer the relation where the
participation is total to avoid null values
 Many to one is represented as foreign key in the Many relation
 Many to Many is represented as two foreign keys for each of the
two relations
Database System Concepts
3.36
©Silberschatz, Korth and Sudarshan
Mapping ER to Relational and Vice-versa
 ER to Relational – see slides in Chp 2
 Relational to ER. Must determine for each foreign key which type
of relationship it represents (may not be unique) or existence of a
weak entity set.
Database System Concepts
3.37
©Silberschatz, Korth and Sudarshan
Schema Diagram for the Banking Enterprise
(another style to denote foreign keys…)
Database System Concepts
3.38
©Silberschatz, Korth and Sudarshan
Database System Concepts
3.39
©Silberschatz, Korth and Sudarshan
Possible relational database state
corresponding to the COMPANY scheme
Database System Concepts
3.40
©Silberschatz, Korth and Sudarshan
SQL -‫הגדרות בסיס נתונים ב‬
CREATE TABLE EMPOLYEE
(
FNAME
VARCHAR(15) NOT NULL.
MINIT
CHAR.
LNAME
VARCHAR(15) NOT NULL.
SSN
CHAR(9)
BDATE
DATE
ADDRESS
VARCHAR(30).
SEX
CHAR.
SALARY
DECIMAL(10,2).
SUPERSSN
CHAR(9).
DNO
INT
NOT NULL.
NOT NULL.
PRIMARY KEY (SSN).
FOREIGN KEY (SUPERSSN) REFERENCES EMPLOYEE (SSN),
FOREIGN KEY (DNO) REFERENCES DEPARTMENT (DNUMBER));
CREATE TABLE DEPARTMENT
(
DNAME
VARCHAR(15) NOT NULL
DNUMBER
INT
NOT NULL
MGRSSN
CHR(9)
NOT NULL
MGRSTARTDATE
DATE,
PRIMARY KEY (DNUMBER)
UNIQUE (DNAME)
FOREIGN KEY (MGRSSN) REFERENCES EMPLOYEE (SSN));
CREATE TABLE DEPT_LOCATIONS
(
DNUMBER
DLOCATION
INT
NOT NULL,
VARCHAR(15) NOT NULL,
PRIMARY KEY (DNUMBER, DLOCATION),
FOREIGN KEY (DNUMBER) REFERENCES DEPARTMENT (DNUMBER)
Database System Concepts
3.41
©Silberschatz, Korth and Sudarshan
SQL -‫הגדרות בסיס נתונים ב‬
CREATE TABLE PROJECT
(
PNAME
VARCHAR(15)
NOT NULL,
PNUMBER
INT
NOT NULL,
PLOCATION
VARCHAR(15) .
DNUM
INT
NOT NULL,
PRIMARY KEY (PNUMBER)
UNIQUE (PNAME)
FOREIGN KEY (DNUM) REFERENCES DEPARTMENT (DNUMBER) );
CREATE TABLE WORKS_ON
(
ESSN
CHAR(9)
NOT NULL,
PNO
INT
NOT NULL,
HOURS
DECIMAL(3, 1)
NOT NULL,
PRIMARY KEY (ESSN, PNO),
FOREIGN KEY (ESSN) REFERENCES EMPLOYEE (SSN),
FOREIGN KEY (PNO) REFERENCES PROJECT (PNUMBER) );
CREATE TABLE DEPENDENT
(
ESSN
CHAR(9)
NOT NULL,
DEPENDENT_NAME
VARCHR(15)NOT NULL,
SEX
CHAR,
BDATE
DATE,
RELATIONSHIP
VARCHAR(8),
PRIMARY KEY (ESSN, DEPENDENR_NAME),
FOREIGN KEY (ESSN) REFERENCES EMPLOYEE (SSN) );
Database System Concepts
3.42
©Silberschatz, Korth and Sudarshan
Query Languages
 Language in which user requests information from the database.
 Categories of languages
• procedural
• non-procedural
 “Pure” languages:
• Relational Algebra
• Tuple Relational Calculus
• Domain Relational Calculus
 Pure languages form underlying basis of query languages that
people use.
Database System Concepts
3.43
©Silberschatz, Korth and Sudarshan
Relational Algebra
 Procedural language
 Six basic operators
• select
• project
• union
• set difference
• Cartesian product
• rename
 The operators take one, two or more relations as inputs and give
a new relation as a result.
Database System Concepts
3.44
©Silberschatz, Korth and Sudarshan
‫אלגברה טבלאית‬
B
A, B, C
project – ‫ הטלה‬
AxB
– ‫ מכפלה קרטזית‬
U
Union – ‫ איחוד‬
∩
Intersection – ‫ חיתוך‬
-
Difference – ‫ הפרש‬
JOIN - ‫ צימוד‬
B
%
Database System Concepts
select – ‫ בחירה‬
3.45
Division – ‫ חילוק‬
©Silberschatz, Korth and Sudarshan
Select Operation – Example
• Relation r
• A=B ^ D > 5 (r)
Database System Concepts
A
B
C
D


1
7


5
7


12
3


23 10
A
B
C
D


1
7


23 10
3.46
©Silberschatz, Korth and Sudarshan
Select Operation
 Notation:
 p(r)
 p is called the selection predicate
 Defined as:
p(r) = {t | t  r and p(t)}
Where p is a formula in propositional calculus consisting
of terms connected by :  (and),  (or),  (not)
Each term is one of:
<attribute> op <attribute> or <constant>
where op is one of: =, , >, . <. 
 Example of selection:
 branch-name=“Perryridge”(account)
Database System Concepts
3.47
©Silberschatz, Korth and Sudarshan
Project Operation – Example
 Relation r:
 A,C (r)
Database System Concepts
A
B
C

10
1

20
1

30
1

40
2
A
C
A
C

1

1

1

1

1

2

2
=
3.48
©Silberschatz, Korth and Sudarshan
Project Operation
 Notation:
A1, A2, …, Ak (r)
where A1, A2 are attribute names and r is a relation name.
 The result is defined as the relation of k columns obtained by
erasing the columns that are not listed
 Duplicate rows removed from result, since relations are sets
 E.g. To eliminate the branch-name attribute of account
account-number, balance (account)
Database System Concepts
3.49
©Silberschatz, Korth and Sudarshan
Union Operation – Example
 Relations r, s:
A
B
A
B

1

2

2

3

1
s
r
r  s:
Database System Concepts
A
B

1

2

1

3
3.50
©Silberschatz, Korth and Sudarshan
Union Operation
 Notation: r  s
 Defined as:
r  s = {t | t  r or t  s}
 For r  s to be valid.
1. r, s must have the same arity (same number of attributes)
2. The attribute domains must be compatible (e.g., 2nd column
of r deals with the same type of values as does the 2nd
column of s, (i.e. usually the same schema ))
 E.g. to find all customers with either an account or a loan
customer-name (depositor)  customer-name (borrower)
Database System Concepts
3.51
©Silberschatz, Korth and Sudarshan
Set Difference Operation – Example
 Relations r, s:
A
B
A
B

1

2

2

3

1
s
r
r – s:
Database System Concepts
A
B

1

1
3.52
©Silberschatz, Korth and Sudarshan
Set Difference Operation
 Notation r – s
 Defined as:
r – s = {t | t  r and t  s}
 Set differences must be taken between compatible relations.
• r and s must have the same arity
• attribute domains of r and s must be compatible
Database System Concepts
3.53
©Silberschatz, Korth and Sudarshan
Set-Intersection Operation
 Notation: r  s
 Defined as:
 r  s ={ t | t  r and t  s }
 Assume:
• r, s have the same arity
• attributes of r and s are compatible
 Note: r  s = r - (r - s)
Database System Concepts
3.54
©Silberschatz, Korth and Sudarshan
Set-Intersection Operation - Example
 Relation r, s:
A
B



1
2
1
A


r
 rs
Database System Concepts
A
B

2
B
2
3
s
3.55
©Silberschatz, Korth and Sudarshan
Database System Concepts
3.56
©Silberschatz, Korth and Sudarshan
Possible relational database state
corresponding to the COMPANY scheme
Database System Concepts
3.57
©Silberschatz, Korth and Sudarshan
Database System Concepts
3.58
©Silberschatz, Korth and Sudarshan
Database System Concepts
3.59
©Silberschatz, Korth and Sudarshan
∩
Database System Concepts
3.60
©Silberschatz, Korth and Sudarshan
Cartesian-Product Operation-Example
Relations r, s:
A
B
C
D
E

1

2




10
10
20
10
a
a
b
b
r
s
r x s:
Database System Concepts
A
B
C
D
E








1
1
1
1
2
2
2
2








10
10
20
10
10
10
20
10
a
a
b
b
a
a
b
b
3.61
©Silberschatz, Korth and Sudarshan
Cartesian-Product Operation
 Notation r x s
 Defined as:
r x s = {t q | t  r and q  s}
 Assume that attributes of r(R) and s(S) are disjoint. (That is, R  S =
).
 If attributes of r(R) and s(S) are not disjoint, then renaming must be
used.
Database System Concepts
3.62
©Silberschatz, Korth and Sudarshan
Rename Operation
 Allows us to name, and therefore to refer to, the results of
relational-algebra expressions.
 Allows us to refer to a relation by more than one name.
Example:
 x (E)
returns the expression E under the name X
If a relational-algebra expression E has arity n, then
x (A1, A2, …, An) (E)
returns the result of expression E under the name X, and with the
attributes renamed to A1, A2, …., An.
Database System Concepts
3.63
©Silberschatz, Korth and Sudarshan
Composition of Operations
 Can build expressions using multiple operations
 Example: A=C(r x s)
 rxs
 A=C(r x s)
Database System Concepts
A
B
C
D
E








1
1
1
1
2
2
2
2








10
10
20
10
10
10
20
10
a
a
b
b
a
a
b
b
A
B
C
D
E



1
2
2
 10
 20
 20
a
a
b
3.64
©Silberschatz, Korth and Sudarshan
Join or Theta Join
Selection over a cartesian product
R
BS
 B (RxS)
Meaning:
For every row r of R
output all rows s of S
which satisfy condition B.
Database System Concepts
3.65
©Silberschatz, Korth and Sudarshan
Natural-Join Operation
n
Notation: r
s
 Let r and s be relations on schemas R and S respectively.
Then, r
s is a relation on schema R  S obtained as follows:
• Consider each pair of tuples tr from r and ts from s.
• If tr and ts have the same value on each of the attributes in R  S, add
a tuple t to the result, where
 t has the same value as t on r
r
 t has the same value as t
s on s
 Example:
R = (A, B, C, D)
S = (E, B, D)
• Result schema = (A, B, C, D, E)
• r
s is defined as:
r.A, r.B, r.C, r.D, s.E (r.B = s.B  r.D = s.D (r x s))
Database System Concepts
3.66
©Silberschatz, Korth and Sudarshan
Natural Join Operation – Example
 Relations r, s:
A
B
C
D
B
D
E





1
2
4
1
2





a
a
b
a
b
1
3
1
2
3
a
a
a
b
b





r
r
s
Database System Concepts
s
A
B
C
D
E





1
1
1
1
2





a
a
a
a
b





3.67
©Silberschatz, Korth and Sudarshan
Natural-Join Operation – my definition
n
Notation: r
s also
r*s
 Let r and s be relations on schemas R and S respectively.
Then, r
s is a relation on schema R  S obtained as follows:
• r and s are joined by some Equi-join
• The redundant (duplicate) attributes are removed
 Example:
R = (A, B, C, D)
S = (E, B, D)
• The equi-join may be on B only
• Examples Dept natural join Emp on Emp-id
•
Dept natural join Emp on Mgr-id
Importance: natural joins along foreign key express Relationship!
To avoid confusion: write the predicate B explicitly!
Database System Concepts
3.68
©Silberschatz, Korth and Sudarshan
Possible relational database state
corresponding to the COMPANY scheme
Database System Concepts
3.69
©Silberschatz, Korth and Sudarshan
Database System Concepts
3.70
©Silberschatz, Korth and Sudarshan
Database System Concepts
3.71
©Silberschatz, Korth and Sudarshan
Division Operation
rs
 Suited to queries that include the phrase “for all”.
 Let r and s be relations on schemas R and S respectively
where
• R = (A1, …, Am, B1, …, Bn)
• S = (B1, …, Bn)
The result of r  s is a relation on schema
R – S = (A1, …, Am)
r  s = { t | t   R-S(r)   u  s ( tu  r ) }
Database System Concepts
3.72
©Silberschatz, Korth and Sudarshan
Division Operation – Example
Relations r, s:
r  s:
A
A
B
B











1
2
3
1
1
1
3
4
6
1
2
1
2
s
r


Database System Concepts
3.73
©Silberschatz, Korth and Sudarshan
Another Division Example
Relations r, s:
A
B
C
D
E
D
E








a
a
a
a
a
a
a
a








a
a
b
a
b
a
b
b
1
1
1
1
3
1
1
1
a
b
1
1
s
r
r  s:
Database System Concepts
A
B
C


a
a


3.74
©Silberschatz, Korth and Sudarshan
Division Operation (Cont.)
 Property
• Let q = r  s
• Then q is the largest relation satisfying q x s  r
 Definition in terms of the basic algebra operation
Let r(R) and s(S) be relations, and let S  R
r  s = R-S (r) –R-S ( (R-S (r) x s) – R-S,S(r))
To see why
• R-S,S(r) simply reorders attributes of r
• T = R-S(R-S (r) x s) – R-S,S(r)) gives those tuples t in
R-S (r) such that for some tuple u  s, tu  r.
• Therefore R-S (r) - T is what we need!
Database System Concepts
3.75
©Silberschatz, Korth and Sudarshan
Illustrating the division operation
(a)Dividing SSN_PNOS by SMITH_PNOS.
(b) T <= R / S
Database System Concepts
3.76
©Silberschatz, Korth and Sudarshan
Banking Example
branch (branch-name, branch-city, assets)
customer (customer-name, customer-street, customer-only)
account (account-number, branch-name, balance)
loan (loan-number, branch-name, amount)
depositor (customer-name, account-number)
borrower (customer-name, loan-number)
Database System Concepts
3.77
©Silberschatz, Korth and Sudarshan
Example Queries
 Find all loans of over $1200
amount > 1200 (loan)
 Find the loan number for each loan of an amount greater than
$1200
loan-number (amount > 1200 (loan))
Database System Concepts
3.78
©Silberschatz, Korth and Sudarshan
Example Queries
 Find the names of all customers who have a loan, an account, or both, from
the bank
customer-name (borrower)  customer-name (depositor)
Find the names of all customers who have a loan and an account at
bank.
customer-name (borrower)  customer-name (depositor)
Database System Concepts
3.79
©Silberschatz, Korth and Sudarshan
Example Queries
 Find the names of all customers who have a loan at the Perryridge
branch.
customer-name (branch-name=“Perryridge”
(borrower.loan-number = loan.loan-number(borrower x loan)))
 Find the names of all customers who have a loan at the Perryridge
branch but do not have an account at any branch of the bank.
customer-name (branch-name = “Perryridge”
(borrower.loan-number = loan.loan-number(borrower x loan))) –
customer-name(depositor)
Database System Concepts
3.80
©Silberschatz, Korth and Sudarshan
Example Queries
 Find the names of all customers who have a loan at the Perryridge
branch.
Query 1
customer-name(branch-name = “Perryridge” (
borrower.loan-number = loan.loan-number(borrower x loan)))
 Query 2
customer-name(loan.loan-number = borrower.loan-number(
(branch-name = “Perryridge”(loan)) x borrower))
Which one is more efficient?
Database System Concepts
3.81
©Silberschatz, Korth and Sudarshan
Example Queries
Find the largest account balance
 Rename account relation as d
 The query is:
balance(account) - account.balance
(account.balance < d.balance (account x d (account)))
Second term is all those accounts which are smaller than some
account…
Database System Concepts
3.82
©Silberschatz, Korth and Sudarshan
Assignment Operation
 The assignment operation () provides a convenient way to express
complex queries.
•
Write query as a sequential program consisting of
 a series of assignments
 followed by an expression whose value is displayed as a result of the
query.
• Assignment must always be made to a temporary relation variable.
 Example: Write r  s as
temp1  R-S (r)
temp2  R-S ((temp1 x s) – R-S,S (r))
result = temp1 – temp2
• The result to the right of the  is assigned to the relation variable on the left of
the .
• May use variable in subsequent expressions.
Database System Concepts
3.83
©Silberschatz, Korth and Sudarshan
Example Queries
 Find all customers who have an account from at least the
“Downtown” and the Uptown” branches.
Query 1
CN(BN=“Downtown”(depositor
CN(BN=“Uptown”(depositor
account)) 
account))
where CN denotes customer-name and BN denotes
branch-name.
Query 2 – using division
customer-name, branch-name (depositor
account)
 temp(branch-name) ({(“Downtown”), (“Uptown”)})
Database System Concepts
3.84
©Silberschatz, Korth and Sudarshan
Example Queries
 Find all customers who have an account at all branches located
in Brooklyn city.
customer-name, branch-name (depositor
account)
 branch-name (branch-city = “Brooklyn” (branch))
Note the right project before the division!
Database System Concepts
3.85
©Silberschatz, Korth and Sudarshan
Extended Relational-Algebra-Operations
 Generalized Projection
 Outer Join
 Aggregate Functions
Database System Concepts
3.86
©Silberschatz, Korth and Sudarshan
Generalized Projection
 Extends the projection operation by allowing arithmetic functions
to be used in the projection list.
 F1, F2, …, Fn(E)
 E is any relational-algebra expression
 Each of F1, F2, …, Fn are are arithmetic expressions involving
constants and attributes in the schema of E.
 Given relation credit-info(customer-name, limit, credit-balance),
find how much more each person can spend:
customer-name, limit – credit-balance (credit-info)
Database System Concepts
3.87
©Silberschatz, Korth and Sudarshan
Aggregate Functions and Operations
 Aggregation function takes a collection of values and returns a
single value as a result.
avg: average value
min: minimum value
max: maximum value
sum: sum of values
count: number of values
 Aggregate operation in relational algebra
G1, G2, …, Gn
g F1( A1), F2( A2),…, Fn( An) (E)
• E is any relational-algebra expression
• G1, G2 …, Gn is a list of attributes on which to group (can be empty)
• Each Fi is an aggregate function
• Each Ai is an attribute name
Database System Concepts
3.88
©Silberschatz, Korth and Sudarshan
Aggregate Operation – Example
 Relation r:
g sum(c) (r)
Database System Concepts
A
B
C








7
7
3
10
sum-C
27
3.89
©Silberschatz, Korth and Sudarshan
Aggregate Operation – Example
 Relation account grouped by branch-name:
branch-name account-number
Perryridge
Perryridge
Brighton
Brighton
Redwood
branch-name
g
A-102
A-201
A-217
A-215
A-222
sum(balance)
400
900
750
750
700
(account)
branch-name
Perryridge
Brighton
Redwood
Database System Concepts
balance
3.90
balance
1300
1500
700
©Silberschatz, Korth and Sudarshan
Aggregate Functions (Cont.)
 Result of aggregation does not have a name
• Can use rename operation to give it a name
• For convenience, we permit renaming as part of aggregate
operation
branch-name
g
sum(balance) as sum-balance (account)
Note: branch-name is the Group-by attribute
sum is the function
balance is the attribute on which the
function operates
account is the relation expression
Database System Concepts
3.91
©Silberschatz, Korth and Sudarshan
Database System Concepts
3.92
©Silberschatz, Korth and Sudarshan
Outer Join
 An extension of the join operation that avoids loss of information.
 Computes the join and then adds tuples form one relation that do
not match tuples in the other relation to the result of the join.
 Uses null values:
• null signifies that the value is unknown or does not exist
• All comparisons involving null are (roughly speaking) false by
definition.
 Will study precise meaning of comparisons with nulls later
Database System Concepts
3.93
©Silberschatz, Korth and Sudarshan
Outer Join – Example
 Relation loan
loan-number
branch-name
L-170
L-230
L-260
Downtown
Redwood
Perryridge
amount
3000
4000
1700
 Relation borrower
customer-name loan-number
Jones
Smith
Hayes
Database System Concepts
L-170
L-230
L-155
3.94
©Silberschatz, Korth and Sudarshan
Outer Join – Example
 Inner Join
loan
Borrower
loan-number
L-170
L-230
branch-name
Downtown
Redwood
amount
customer-name
3000
4000
Jones
Smith
amount
customer-name
 Left Outer Join
loan
Borrower
loan-number
L-170
L-230
L-260
Database System Concepts
branch-name
Downtown
Redwood
Perryridge
3000
4000
1700
3.95
Jones
Smith
null
©Silberschatz, Korth and Sudarshan
Outer Join – Example
 Right Outer Join
loan
borrower
loan-number
L-170
L-230
L-155
branch-name
Downtown
Redwood
null
amount
3000
4000
null
customer-name
Jones
Smith
Hayes
 Full Outer Join
loan
borrower
loan-number
L-170
L-230
L-260
L-155
Database System Concepts
branch-name
Downtown
Redwood
Perryridge
null
amount
3000
4000
1700
null
3.96
customer-name
Jones
Smith
null
Hayes
©Silberschatz, Korth and Sudarshan
A two level recursive query
Database System Concepts
3.97
©Silberschatz, Korth and Sudarshan
Null Values
 It is possible for tuples to have a null value, denoted by null, for
some of their attributes
 null signifies an unknown value or that a value does not exist.
 The result of any arithmetic expression involving null is null.
 Aggregate functions simply ignore null values
• Is an arbitrary decision. Could have returned null as result instead.
• We follow the semantics of SQL in its handling of null values
 For duplicate elimination and grouping, null is treated like any
other value, and two nulls are assumed to be the same
• Alternative: assume each null is different from each other
• Both are arbitrary decisions, so we simply follow SQL
Database System Concepts
3.98
©Silberschatz, Korth and Sudarshan
Null Values
 Comparisons with null values return the special truth value unknown
• If false was used instead of unknown, then
would not be equivalent to
not (A < 5)
A >= 5
 Three-valued logic using the truth value unknown:
• OR: (unknown or true)
= true,
(unknown or false)
= unknown
(unknown or unknown) = unknown
• AND: (true and unknown)
= unknown,
(false and unknown)
= false,
(unknown and unknown) = unknown
• NOT: (not unknown) = unknown
• In SQL “P is unknown” evaluates to true if predicate P evaluates to
unknown
 Result of select predicate is treated as false if it evaluates to
unknown
Database System Concepts
3.99
©Silberschatz, Korth and Sudarshan
Modification of the Database
 The content of the database may be modified using the following
operations:
• Deletion
• Insertion
• Updating
 All these operations are expressed using the assignment
operator.
Database System Concepts
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©Silberschatz, Korth and Sudarshan
Deletion
 A delete request is expressed similarly to a query, except instead
of displaying tuples to the user, the selected tuples are removed
from the database.
 Can delete only whole tuples; cannot delete values on only
particular attributes
 A deletion is expressed in relational algebra by:
rr–E
where r is a relation and E is a relational algebra query.
Database System Concepts
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©Silberschatz, Korth and Sudarshan
Deletion Examples
 Delete all account records in the Perryridge branch.
account  account – branch-name = “Perryridge” (account)
 Delete all loan records with amount in the range of 0 to 50
loan  loan –  amount 0 and amount  50 (loan)
 Delete all accounts at branches located in Needham.
r1   branch-city = “Needham” (account
branch)
r2  branch-name, account-number, balance (r1)
r3   customer-name, account-number (r2
depositor)
account  account – r2
depositor  depositor – r3
Database System Concepts
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©Silberschatz, Korth and Sudarshan
Insertion
 To insert data into a relation, we either:
• specify a tuple to be inserted
• write a query whose result is a set of tuples to be inserted
 in relational algebra, an insertion is expressed by:
r r  E
where r is a relation and E is a relational algebra expression.
 The insertion of a single tuple is expressed by letting E be a
constant relation containing one tuple.
Database System Concepts
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©Silberschatz, Korth and Sudarshan
Insertion Examples
 Insert information in the database specifying that Smith has
$1200 in account A-973 at the Perryridge branch.
account  account  {(“Perryridge”, A-973, 1200)}
depositor  depositor  {(“Smith”, A-973)}
• Provide as a gift for all loan customers in the Perryridge
branch, a $200 savings account. Let the loan number serve
as the account number for the new savings account.
r1  (branch-name = “Perryridge” (borrower
loan))
account  account  branch-name, account-number,200 (r1)
depositor  depositor  customer-name, loan-number(r1)
Database System Concepts
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©Silberschatz, Korth and Sudarshan
Updating
 A mechanism to change a value in a tuple without charging all
values in the tuple
 Use the generalized projection operator to do this task
r   F1, F2, …, FI, (r)
 Each Fi is either
• the ith attribute of r, if the ith attribute is not updated, or,
• if the attribute is to be updated Fi is an expression, involving only
constants and the attributes of r, which gives the new value for the
attribute
Database System Concepts
3.105
©Silberschatz, Korth and Sudarshan
Update Examples
 Make interest payments by increasing all balances by 5 percent.
account   AN, BN, BAL * 1.05 (account)
where AN, BN and BAL stand for account-number, branch-name
and balance, respectively.
 Pay all accounts with balances over $10,000 6 percent interest
and pay all others 5 percent
account 
Database System Concepts
 AN, BN, BAL * 1.06 ( BAL  10000 (account))
 AN, BN, BAL * 1.05 (BAL  10000 (account))
3.106
©Silberschatz, Korth and Sudarshan
Summary – operations of the relational algebra
Operation
Notation
Purpose
SELECT
Selects all tuples that satisfy the selection  < selection condition> (R)
condition from a relative R.
PROJECT
Produces a new relation with only some
of the attributes of R, and removes
duplicate tuples.
 < attribute list > (R)
THETA JOIN
Produces all combinations of tuples from
R1 and R2 that satisfy the join condition.
R1 < join condition > R2
EQUIJOIN
Produces all the combinations of tuples
from R1 and R2 that satisfy a join
condition with only equality
comparisons.
R1 < join condition > R2, or
R1 (< join attributes 1>),
(<join attributes 2>R2
NATURAL JOIN
Same as equijoin except that the join
attributes of R2 are not included in the
resulting relation; (note difference with
[S])
R1 < join condition > R2, or
R1 (< join attributes 1>),
(<join attributes 2>)
R2 or R1*R2
Database System Concepts
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©Silberschatz, Korth and Sudarshan
Summary – operations of the relational algebra – cont.
Operation
UNION
INTERSECTION
Purpose
Products a relation that includes all the tuples
in R1 or R2 or both R1 or R2; R1 and R2 must
be union compatible.
Produces a relation that includes all the
tuples in R1 or R2 or both R1 and R2;
R1 and R2 must be union compatible.
Notation
R1  R2
R1  R2
R1 – R2
DIFFERENCE
Produces a relation that includes all the
tuples in R1 that are not in R2;
.
CARTESIAN PRODUCT
DIVISION
Database System Concepts
Produces a relation that has the attributes of
R1 and R2 and includes as tuples all possible
combinations of tuples from R1 and R2.
R 1 X R2
Produces a relation R(X) that includes all
tuples t[χ] in that appear in R1 in combination
with every tuple from R2(Y), where Z = X 
Y.
R1(Z) ÷ R2(Y)
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©Silberschatz, Korth and Sudarshan
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
 t  r denotes that tuple t is in relation r
 P is a formula similar to that of the predicate calculus
Database System Concepts
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©Silberschatz, Korth and Sudarshan
Predicate Calculus Formula
1. Set of attributes and constants
2. Set of comparison operators: (e.g., , , , , , )
3. Set of connectives: and (), or (v)‚ not ()
4. Implication (): x  y, if x if true, then y is true
x  y x v y
5. Set of quantifiers:

 t  r (Q(t))  ”there exists” a tuple in t in relation r
such that predicate Q(t) is true

t r (Q(t)) Q is true “for all” tuples t in relation r
Database System Concepts
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©Silberschatz, Korth and Sudarshan
A Valid TRC – my definition
{t1.A, t2 .B,… tn .Z | P (t1, t2, …,tn , tn+1, …,tm) }

t1.A, t2 .B,… tn .Z are tuple variables which define the output.
each must be defined over a single relation, they must remain free in P, i.e
not associated with quantifiers

tn+1, …,tm are tuple variables which must be defined over relations, and must
be bound by a quantifier.

Semantics: run the free t’s on all their corresponding relations, and for each
combination, check whether the P is true, if it is, output the defined output
values.

A variable is defined over a relation either as: t R or R(t), both syntaxes are
ok and will be used.

Value of a variable may be defined as t.A or t[A], both syntaxes are ok.
Database System Concepts
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©Silberschatz, Korth and Sudarshan