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Database System Concepts, 6th Ed. ©Silberschatz, Korth and Sudarshan See www.db-book.com for conditions on re-use

Chapter 6: Formal Relational Query Languages

©Silberschatz, Korth and Sudarshan

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Database System Concepts - 6th Edition

Chapter 6: Formal Relational Query Languages

Relational Algebra
Tuple Relational Calculus
Domain Relational Calculus

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Database System Concepts - 6th Edition

Relational Algebra

Procedural language
Six basic operators
select: 
project: 
union: 
set difference: –
Cartesian product: x
rename: 
The operators take one or two relations as inputs and produce a new relation as a result.

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Database System Concepts - 6th Edition

Select Operation – Example

Relation r

A=B ^ D > 5 (r)

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Database System Concepts - 6th Edition

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:

where op is one of: =, , >, . <. 

Example of selection:

 dept_name=“Physics”(instructor)

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Database System Concepts - 6th Edition

Project Operation – Example

Relation r:

A,C (r)

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Database System Concepts - 6th Edition

Project Operation

Notation:

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
Example: To eliminate the dept_name attribute of instructor

ID, name, salary (instructor)

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Database System Concepts - 6th Edition

Union Operation – Example

Relations r, s:

r  s:

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Database System Concepts - 6th Edition

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 (example: 2nd column
of r deals with the same type of values as does the 2nd
column of s)

Example: to find all courses taught in the Fall 2009 semester, or in the Spring 2010 semester, or in both
course_id ( semester=“Fall” Λ year=2009 (section)) 
course_id ( semester=“Spring” Λ year=2010 (section))

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Database System Concepts - 6th Edition

Set difference of two relations

Relations r, s:

r – s:

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Database System Concepts - 6th Edition

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
Example: to find all courses taught in the Fall 2009 semester, but not in the Spring 2010 semester
course_id ( semester=“Fall” Λ year=2009 (section)) −
course_id ( semester=“Spring” Λ year=2010 (section))

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Database System Concepts - 6th Edition

Cartesian-Product Operation – Example

Relations r, s:

r x s:

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Database System Concepts - 6th Edition

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.

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Database System Concepts - 6th Edition

Composition of Operations

Can build expressions using multiple operations
Example: A=C(r x s)

r x s

A=C(r x s)

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Database System Concepts - 6th Edition

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

returns the result of expression E under the name X, and with the

attributes renamed to A1 , A2 , …., An .

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Database System Concepts - 6th Edition

Example Query

Find the largest salary in the university
Step 1: find instructor salaries that are less than some other instructor salary (i.e. not maximum)

using a copy of instructor under a new name d

instructor.salary ( instructor.salary < d,salary
(instructor x d (instructor)))
Step 2: Find the largest salary
salary (instructor) –
instructor.salary ( instructor.salary < d,salary
(instructor x d (instructor)))

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Database System Concepts - 6th Edition

Example Queries

Find the names of all instructors in the Physics department, along with the course_id of all courses they have taught

Query 1
instructor.ID,course_id (dept_name=“Physics” (
 instructor.ID=teaches.ID (instructor x teaches)))

Query 2
instructor.ID,course_id (instructor.ID=teaches.ID (
 dept_name=“Physics” (instructor) x teaches))

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Database System Concepts - 6th Edition

Formal Definition

A basic expression in the relational algebra consists of either one of the following:
A relation in the database
A constant relation
Let E1 and E2 be relational-algebra expressions; the following are all relational-algebra expressions:
E1  E2
E1 – E2
E1 x E2
p (E1), P is a predicate on attributes in E1
s(E1), S is a list consisting of some of the attributes in E1
 x (E1), x is the new name for the result of E1

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Database System Concepts - 6th Edition

Additional Operations

We define additional operations that do not add any power to the

relational algebra, but that simplify common queries.

Set intersection
Natural join
Assignment
Outer join

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Database System Concepts - 6th Edition

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)

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Database System Concepts - 6th Edition

Set-Intersection Operation – Example

Relation r, s:

r  s

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Database System Concepts - 6th Edition

Natural-Join Operation

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 tr on r
t has the same value as ts 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))

Notation: r s

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Database System Concepts - 6th Edition

Natural Join Example

Relations r, s:

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Database System Concepts - 6th Edition

Natural Join and Theta Join

Find the names of all instructors in the Comp. Sci. department together with the course titles of all the courses that the instructors teach
 name, title ( dept_name=“Comp. Sci.” (instructor teaches course))
Natural join is associative
(instructor teaches) course is equivalent to
instructor (teaches course)
Natural join is commutative
instruct teaches is equivalent to
teaches instructor
The theta join operation r  s is defined as
r  s =  (r x s)

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Database System Concepts - 6th Edition

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.

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Database System Concepts - 6th Edition

Outer Join

An extension of the join operation that avoids loss of information.
Computes the join and then adds tuples form one relation that does 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.
We shall study precise meaning of comparisons with nulls later

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Database System Concepts - 6th Edition

Outer Join – Example

Relation instructor1

Relation teaches1

course_id

10101

12121

76766

CS-101

FIN-201

BIO-101

Comp. Sci.

Finance

Music

dept_name

10101

12121

15151

name

Srinivasan

Mozart

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Database System Concepts - 6th Edition

Outer Join – Example

Join

instructor teaches

Left Outer Join

instructor teaches

dept_name

10101

12121

Comp. Sci.

Finance

course_id

CS-101

FIN-201

name

Srinivasan

dept_name

10101

12121

15151

Comp. Sci.

Finance

Music

course_id

CS-101

FIN-201

null

name

Srinivasan

Mozart

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Database System Concepts - 6th Edition

Outer Join – Example

Full Outer Join

instructor teaches

Right Outer Join

instructor teaches

dept_name

10101

12121

76766

Comp. Sci.

Finance

null

course_id

CS-101

FIN-201

BIO-101

name

Srinivasan

null

dept_name

10101

12121

15151

76766

Comp. Sci.

Finance

Music

null

course_id

CS-101

FIN-201

null

BIO-101

name

Srinivasan

Mozart

null

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Database System Concepts - 6th Edition

Outer Join using Joins

Outer join can be expressed using basic operations
e.g. r s can be written as

(r s) U (r – ∏R(r s) x {(null, …, null)}

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Database System Concepts - 6th Edition

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 (as in SQL)
For duplicate elimination and grouping, null is treated like any other value, and two nulls are assumed to be the same (as in SQL)

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Database System Concepts - 6th Edition

Null Values

Comparisons with null values return the special truth value: unknown
If false was used instead of unknown, then not (A < 5)
would not be equivalent to 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

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Database System Concepts - 6th Edition

Division Operator

Given relations r(R) and s(S), such that S  R, r  s is the largest relation t(R-S) such that
t x s  r
E.g. let r(ID, course_id) = ID, course_id (takes ) and
s(course_id) = course_id (dept_name=“Biology”(course )
then r  s gives us students who have taken all courses in the Biology department
Can 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.

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Database System Concepts - 6th Edition

Extended Relational-Algebra-Operations

Generalized Projection
Aggregate Functions

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Database System Concepts - 6th Edition

Generalized Projection

Extends the projection operation by allowing arithmetic functions to be used in the projection list.
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 instructor(ID, name, dept_name, salary) where salary is annual salary, get the same information but with monthly salary

ID, name, dept_name, salary/12 (instructor)

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Database System Concepts - 6th Edition

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

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
Note: Some books/articles use  instead of (Calligraphic G)

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Database System Concepts - 6th Edition

Aggregate Operation – Example

Relation r:









sum(c) (r)

sum(c )

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Database System Concepts - 6th Edition

Aggregate Operation – Example

Find the average salary in each department

dept_name avg(salary) (instructor)

avg_salary

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Database System Concepts - 6th Edition

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

dept_name avg(salary) as avg_sal (instructor)

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Database System Concepts - 6th Edition

Modification of the Database

The content of the database may be modified using the following operations:
Deletion
Insertion
Updating
All these operations can be expressed using the assignment operator

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Database System Concepts - 6th Edition

Multiset Relational Algebra

Pure relational algebra removes all duplicates
e.g. after projection
Multiset relational algebra retains duplicates, to match SQL semantics
SQL duplicate retention was initially for efficiency, but is now a feature
Multiset relational algebra defined as follows
selection: has as many duplicates of a tuple as in the input, if the tuple satisfies the selection
projection: one tuple per input tuple, even if it is a duplicate
cross product: If there are m copies of t1 in r, and n copies of t2 in s, there are m x n copies of t1.t2 in r x s
Other operators similarly defined
E.g. union: m + n copies, intersection: min(m, n) copies
difference: min(0, m – n) copies

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Database System Concepts - 6th Edition

SQL and Relational Algebra

select A1, A2, .. An
from r1, r2, …, rm
where P

is equivalent to the following expression in multiset relational algebra

 A1, .., An ( P (r1 x r2 x .. x rm))

select A1, A2, sum(A3)
from r1, r2, …, rm
where P
group by A1, A2

is equivalent to the following expression in multiset relational algebra

A1, A2 sum(A3) ( P (r1 x r2 x .. x rm)))

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Database System Concepts - 6th Edition

SQL and Relational Algebra

More generally, the non-aggregated attributes in the select clause may be a subset of the group by attributes, in which case the equivalence is as follows:

select A1, sum(A3)
from r1, r2, …, rm
where P
group by A1, A2

is equivalent to the following expression in multiset relational algebra

 A1,sumA3( A1,A2 sum(A3) as sumA3( P (r1 x r2 x .. x rm)))

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Database System Concepts - 6th Edition

Tuple Relational Calculus

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Database System Concepts - 6th Edition

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

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Database System Concepts - 6th Edition

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

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Database System Concepts - 6th Edition

Example Queries

Find the ID, name, dept_name, salary for instructors whose salary is greater than $80,000

As in the previous query, but output only the ID attribute value

{t |  s instructor (t [ID ] = s [ID ]  s [salary ]  80000)}

Notice that a relation on schema (ID) is implicitly defined by

the query

{t | t  instructor  t [salary ]  80000}

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Database System Concepts - 6th Edition

Example Queries

Find the names of all instructors whose department is in the Watson building

{t | s  section (t [course_id ] = s [course_id ] 
s [semester] = “Fall”  s [year] = 2009
v u  section (t [course_id ] = u [course_id ] 
u [semester] = “Spring”  u [year] = 2010)}

Find the set of all courses taught in the Fall 2009 semester, or in
the Spring 2010 semester, or both

{t | s  instructor (t [name ] = s [name ]
 u  department (u [dept_name ] = s[dept_name] “
 u [building] = “Watson” ))}

©Silberschatz, Korth and Sudarshan

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Database System Concepts - 6th Edition

Example Queries

{t | s  section (t [course_id ] = s [course_id ] 
s [semester] = “Fall”  s [year] = 2009
 u  section (t [course_id ] = u [course_id ] 
u [semester] = “Spring”  u [year] = 2010)}

Find the set of all courses taught in the Fall 2009 semester, and in
the Spring 2010 semester

{t | s  section (t [course_id ] = s [course_id ] 
s [semester] = “Fall”  s [year] = 2009
  u  section (t [course_id ] = u [course_id ] 
u [semester] = “Spring”  u [year] = 2010)}

Find the set of all courses taught in the Fall 2009 semester, but not in
the Spring 2010 semester

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Database System Concepts - 6th Edition

Safety of Expressions

It is possible to write tuple calculus expressions that generate infinite relations.
For example, { t |  t r } results in an infinite relation if the domain of any attribute of relation r is infinite
To guard against the problem, we restrict the set of allowable expressions to safe expressions.
An expression {t | P (t )} in the tuple relational calculus is safe if every component of t appears in one of the relations, tuples, or constants that appear in P
NOTE: this is more than just a syntax condition.
E.g. { t | t [A] = 5  true } is not safe --- it defines an infinite set with attribute values that do not appear in any relation or tuples or constants in P.

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Database System Concepts - 6th Edition

Universal Quantification

Find all students who have taken all courses offered in the Biology department
{t |  r  student (t [ID] = r [ID]) 
( u  course (u [dept_name]=“Biology” 
 s  takes (t [ID] = s [ID ] 
s [course_id] = u [course_id]))}
Note that without the existential quantification on student, the above query would be unsafe if the Biology department has not offered any courses.

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Database System Concepts - 6th Edition

Domain Relational Calculus

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Database System Concepts - 6th Edition

Domain Relational Calculus

A nonprocedural query language equivalent in power to the tuple relational calculus
Each query is an expression of the form:

{  x1, x2, …, xn  | P (x1, x2, …, xn)}

x1, x2, …, xn represent domain variables
P represents a formula similar to that of the predicate calculus

©Silberschatz, Korth and Sudarshan

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Database System Concepts - 6th Edition

Example Queries

Find the ID, name, dept_name, salary for instructors whose salary is greater than $80,000
{< i, n, d, s> | < i, n, d, s>  instructor  s  80000}
As in the previous query, but output only the ID attribute value
{< i> | < i, n, d, s>  instructor  s  80000}
Find the names of all instructors whose department is in the Watson building

{< n > |  i, d, s (< i, n, d, s >  instructor
  b, a (< d, b, a>  department  b = “Watson” ))}

©Silberschatz, Korth and Sudarshan

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Database System Concepts - 6th Edition

Example Queries

{<c> |  a, s, y, b, r, t ( <c, a, s, y, b, t >  section 
s = “Fall”  y = 2009 )
v  a, s, y, b, r, t ( <c, a, s, y, b, t >  section ] 
s = “Spring”  y = 2010)}

Find the set of all courses taught in the Fall 2009 semester, or in
the Spring 2010 semester, or both

This case can also be written as
{<c> |  a, s, y, b, r, t ( <c, a, s, y, b, t >  section 
( (s = “Fall”  y = 2009 ) v (s = “Spring”  y = 2010))}

Find the set of all courses taught in the Fall 2009 semester, and in
the Spring 2010 semester

{<c> |  a, s, y, b, r, t ( <c, a, s, y, b, t >  section 
s = “Fall”  y = 2009 )
  a, s, y, b, r, t ( <c, a, s, y, b, t >  section ] 
s = “Spring”  y = 2010)}

©Silberschatz, Korth and Sudarshan

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Database System Concepts - 6th Edition

Safety of Expressions

The expression:

{  x1, x2, …, xn  | P (x1, x2, …, xn )}

is safe if all of the following hold:

All values that appear in tuples of the expression are values from dom (P ) (that is, the values appear either in P or in a tuple of a relation mentioned in P ).

For every “there exists” subformula of the form  x (P1(x )), the subformula is true if and only if there is a value of x in dom (P1) such that P1(x ) is true.

For every “for all” subformula of the form x (P1 (x )), the subformula is true if and only if P1(x ) is true for all values x from dom (P1).

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Database System Concepts - 6th Edition

Universal Quantification

Find all students who have taken all courses offered in the Biology department
{< i > |  n, d, tc ( < i, n, d, tc >  student 
( ci, ti, dn, cr ( < ci, ti, dn, cr >  course  dn =“Biology”
  si, se, y, g ( <i, ci, si, se, y, g>  takes ))}
Note that without the existential quantification on student, the above query would be unsafe if the Biology department has not offered any courses.

* Above query fixes bug in page 246, last query

End of Chapter 6

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Database System Concepts - 6th Edition

Figure 6.01

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Database System Concepts - 6th Edition

Figure 6.02

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Database System Concepts - 6th Edition

Figure 6.03

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Database System Concepts - 6th Edition

Figure 6.04

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Database System Concepts - 6th Edition

Figure 6.05

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Database System Concepts - 6th Edition

Figure 6.06

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Database System Concepts - 6th Edition

Figure 6.07

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Database System Concepts - 6th Edition

Figure 6.08

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Database System Concepts - 6th Edition

Figure 6.09

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Database System Concepts - 6th Edition

Figure 6.10

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Database System Concepts - 6th Edition

Figure 6.11

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Database System Concepts - 6th Edition

Figure 6.12

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Database System Concepts - 6th Edition

Figure 6.13

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Database System Concepts - 6th Edition

Figure 6.14

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Database System Concepts - 6th Edition

Figure 6.15

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Database System Concepts - 6th Edition

Figure 6.16

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Database System Concepts - 6th Edition

Figure 6.17

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Database System Concepts - 6th Edition

Figure 6.18

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Database System Concepts - 6th Edition

Figure 6.19

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Database System Concepts - 6th Edition

Figure 6.20

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Database System Concepts - 6th Edition

Figure 6.21

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Database System Concepts - 6th Edition

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.

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Database System Concepts - 6th Edition

Deletion Examples

Delete all account records in the Perryridge branch.

Delete all accounts at branches located in Needham.

Delete all loan records with amount in the range of 0 to 50

loan  loan – amount 0and amount  50 (loan)

account  account – branch_name = “Perryridge” (account )

r1  branch_city = “Needham” (account branch )

r2   account_number, branch_name, balance (r1)

r3   customer_name, account_number (r2 depositor)

account  account – r2

depositor  depositor – r3

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Database System Concepts - 6th Edition

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.

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Database System Concepts - 6th Edition

Insertion Examples

Insert information in the database specifying that Smith has $1200 in account A-973 at the Perryridge branch.

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.

account  account  {(“A-973”, “Perryridge”, 1200)}

depositor  depositor  {(“Smith”, “A-973”)}

r1  (branch_name = “Perryridge” (borrower loan))

account  account  loan_number, branch_name, 200 (r1)

depositor  depositor  customer_name, loan_number (r1)

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Database System Concepts - 6th Edition

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

Each Fi is either
the I th attribute of r, if the I th 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

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Database System Concepts - 6th Edition

Update Examples

Make interest payments by increasing all balances by 5 percent.

Pay all accounts with balances over $10,000 6 percent interest
and pay all others 5 percent

account   account_number, branch_name, balance * 1.06 ( BAL  10000 (account ))
  account_number, branch_name, balance * 1.05 (BAL  10000 (account))

account   account_number, branch_name, balance * 1.05 (account)

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Database System Concepts - 6th Edition

Example Queries

Find the name of all customers who have a loan at the bank and the loan amount

Find the names of all customers who have a loan and an account at bank.

customer_name (borrower)  customer_name (depositor)

customer_name, loan_number, amount (borrower loan)

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Database System Concepts - 6th Edition

Example Queries

Find all customers who have an account from at least the “Downtown” and the Uptown” branches.

Query 1

customer_name (branch_name = “Downtown” (depositor account )) 

customer_name (branch_name = “Uptown” (depositor account))

Query 2

customer_name, branch_name (depositor account)
 temp(branch_name) ({(“Downtown” ), (“Uptown” )})

Note that Query 2 uses a constant relation.

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Database System Concepts - 6th Edition

Bank 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))

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