Database System Concepts, 6th Ed.
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
Chapter 3: Formal Relational Query
Languages
©Silberschatz, Korth and Sudarshan3.2Database System Concepts - 6th Edition
Chapter 3: Formal Relational Query Languages
Relational Algebra - Extensions
Tuple Relational Calculus
Domain Relational Calculus
©Silberschatz, Korth and Sudarshan3.3Database 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.
©Silberschatz, Korth and Sudarshan3.4Database System Concepts - 6th Edition
Formal Definition
A basic expression in 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 also
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
©Silberschatz, Korth and Sudarshan3.5Database System Concepts - 6th Edition
Additional Operations
We define additional operations that do not add any power to the
relational algebra, but they simplify common queries.
Set intersection
Natural join
Assignment
Outer join
©Silberschatz, Korth and Sudarshan3.6Database 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)
©Silberschatz, Korth and Sudarshan3.7Database System Concepts - 6th Edition
Set-Intersection Operation – Example
Relation r, s:
r  s
©Silberschatz, Korth and Sudarshan3.8Database System Concepts - 6th Edition
Notation: r s
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))
©Silberschatz, Korth and Sudarshan3.9Database System Concepts - 6th Edition
Natural Join Example
Relations r, s:
r s
©Silberschatz, Korth and Sudarshan3.10Database 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)
©Silberschatz, Korth and Sudarshan3.11Database 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.
©Silberschatz, Korth and Sudarshan3.12Database System Concepts - 6th Edition
Outer Join
An extension of the join operation that avoids loss of information.
Computes the join and then adds tuples from 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 the precise meaning of comparisons with nulls
later
©Silberschatz, Korth and Sudarshan3.13Database System Concepts - 6th Edition
Outer Join – Example
Relation instructor1
Relation teaches1
ID course_id
10101
12121
76766
CS-101
FIN-201
BIO-101
Comp. Sci.
Finance
Music
ID dept_name
10101
12121
15151
name
Srinivasan
Wu
Mozart
©Silberschatz, Korth and Sudarshan3.14Database System Concepts - 6th Edition
Left Outer Join
instructor teaches
Outer Join – Example
Join
instructor teaches
ID dept_name
10101
12121
Comp. Sci.
Finance
course_id
CS-101
FIN-201
name
Srinivasan
Wu
ID dept_name
10101
12121
15151
Comp. Sci.
Finance
Music
course_id
CS-101
FIN-201
null
name
Srinivasan
Wu
Mozart
©Silberschatz, Korth and Sudarshan3.15Database System Concepts - 6th Edition
Outer Join – Example
Full Outer Join
instructor teaches
Right Outer Join
instructor teaches
ID dept_name
10101
12121
76766
Comp. Sci.
Finance
null
course_id
CS-101
FIN-201
BIO-101
name
Srinivasan
Wu
null
ID dept_name
10101
12121
15151
76766
Comp. Sci.
Finance
Music
null
course_id
CS-101
FIN-201
null
BIO-101
name
Srinivasan
Wu
Mozart
null
©Silberschatz, Korth and Sudarshan3.16Database 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)}
©Silberschatz, Korth and Sudarshan3.17Database 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
For duplicate elimination and grouping, null is treated like any other
value, and two nulls are assumed to be the same
©Silberschatz, Korth and Sudarshan3.18Database 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 a select predicate is treated as false if it evaluates to
unknown
©Silberschatz, Korth and Sudarshan3.19Database 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 .
If u = r x s than u  r = s division can be seen as invers of cart. prod.
©Silberschatz, Korth and Sudarshan3.20Database System Concepts - 6th Edition
Extended Relational-Algebra-Operations
Generalized Projection
Aggregate Functions
©Silberschatz, Korth and Sudarshan3.21Database 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 is an arithmetic expression 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)
)(,...,, 21
EnFFF
©Silberschatz, Korth and Sudarshan3.22Database 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)
)()(,,(),(,,, 221121
Ennn AFAFAFGGG 
©Silberschatz, Korth and Sudarshan3.23Database System Concepts - 6th Edition
Aggregate Operation – Example
Relation r:
A B








C
7
7
3
10
sum(c) (r) sum(c )
27
©Silberschatz, Korth and Sudarshan3.24Database System Concepts - 6th Edition
Aggregate Operation – Example
Find the average salary in each department
dept_name avg(salary) (instructor)
avg_salary
©Silberschatz, Korth and Sudarshan3.25Database 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 the aggregate
operation
dept_name avg(salary) as avg_sal (instructor)
©Silberschatz, Korth and Sudarshan3.26Database 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
©Silberschatz, Korth and Sudarshan3.27Database 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.
©Silberschatz, Korth and Sudarshan3.28Database System Concepts - 6th Edition
Deletion Examples
Delete all account records in the Perryridge branch.
Delete all accounts at branches located in Needham.
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
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 )
©Silberschatz, Korth and Sudarshan3.29Database 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.
©Silberschatz, Korth and Sudarshan3.30Database 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)
©Silberschatz, Korth and Sudarshan3.31Database 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
)(,,,, 21
rr lFFF 

©Silberschatz, Korth and Sudarshan3.32Database 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)
©Silberschatz, Korth and Sudarshan3.33Database System Concepts - 6th Edition
Example Queries
Find the names of all customers who have a loan and an account at
bank.
customer_name (borrower)  customer_name (depositor)
Find the name of all customers who have a loan at the bank and the
loan amount
customer_name, loan_number, amount (borrower loan)
©Silberschatz, Korth and Sudarshan3.34Database System Concepts - 6th Edition
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.
Example Queries
Find all customers who have an account from at least the “Downtown”
and the Uptown” branches.
©Silberschatz, Korth and Sudarshan3.35Database System Concepts - 6th Edition
Tuple Relational Calculus
©Silberschatz, Korth and Sudarshan3.36Database 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 the relation r
P is a formula similar to that of the predicate calculus
©Silberschatz, Korth and Sudarshan3.37Database 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 t in the relation r
such that predicate Q (t ) is true
 t  r (Q (t ))  Q is true “for all” tuples t in the relation r
©Silberschatz, Korth and Sudarshan3.38Database 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}
©Silberschatz, Korth and Sudarshan3.39Database 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 Sudarshan3.40Database 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.
©Silberschatz, Korth and Sudarshan3.41Database System Concepts - 6th Edition
Domain Relational Calculus
©Silberschatz, Korth and Sudarshan3.42Database 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 Sudarshan3.43Database 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 Sudarshan3.44Database System Concepts - 6th Edition
Example Queries
{<c> |  a, s, y, b, r, t ( <c, a, s, y, b, r, t >  section 
s = “Fall”  y = 2009 )
v  a, s, y, b, r, t ( <c, a, s, y, b, r, 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, r, 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, r, t >  section 
s = “Fall”  y = 2009 )
  a, s, y, b, r, t ( <c, a, s, y, b, r, t >  section ] 
s = “Spring”  y = 2010)}
Database System Concepts, 6th Ed.
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
End of Chapter 3