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Copyright 2007 Ramez Elmasri and Shamkant B.

Navathe
Chapter 3
Relational Model
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 2
Chapter Outline
Relational Model Concepts
Relational Algebra
Unary Relational Operations
Relational Algebra Operations From Set Theory
Binary Relational Operations
Relational Calculus
Tuple Relational Calculus
Domain Relational Calculus

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 3
Relational Model Concepts
A Relation is a mathematical concept based on
the ideas of sets
The model was first proposed by Dr. E.F. Codd of
IBM Research in 1970 in the following paper:
"A Relational Model for Large Shared Data
Banks," Communications of the ACM, June 1970
The above paper caused a major revolution in the
field of database management and earned Dr.
Codd the coveted ACM Turing Award
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 4
Informal Definitions

Informally, a relation looks like a table of values.

A relation typically contains a set of rows.

The data elements in each row represent certain facts that
correspond to a real-world entity or relationship
In the formal model, rows are called tuples

Each column has a column header that gives an indication
of the meaning of the data items in that column
In the formal model, the column header is called an attribute
name (or just attribute)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 5
Example of a Relation
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 6
Informal Definitions
Key of a Relation:
Each row has a value of a data item (or set of items)
that uniquely identifies that row in the table
Called the key
In the STUDENT table, SSN is the key

Sometimes row-ids or sequential numbers are
assigned as keys to identify the rows in a table
Called artificial key or surrogate key

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 7
Formal Definitions - Schema
The Schema (or description) of a Relation:
Denoted by R(A1, A2, .....An)
R is the name of the relation
The attributes of the relation are A1, A2, ..., An
Example:
CUSTOMER is the relation name
Defined over the four attributes: Cust-id, Cust-name,
Each attribute has a domain or a set of valid values.
For example, the domain of Cust-id is 6 digit numbers.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 8
Formal Definitions - Tuple
A tuple is an ordered set of values (enclosed in angled
brackets < >)
Each value is derived from an appropriate domain.
A row in the CUSTOMER relation is a 4-tuple and would
consist of four values, for example:
<632895, "John Smith", "101 Main St. Atlanta, GA 30332",
"(404) 894-2000">
This is called a 4-tuple as it has 4 values
A tuple (row) in the CUSTOMER relation.
A relation is a set of such tuples (rows)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 9
Formal Definitions - Domain
A domain has a logical definition:
Example: USA_phone_numbers are the set of 10 digit phone
numbers valid in the U.S.
A domain also has a data-type or a format defined for it.
The USA_phone_numbers may have a format: (ddd)ddd-dddd where
each d is a decimal digit.
Dates have various formats such as year, month, date formatted
as yyyy-mm-dd, or as dd mm,yyyy etc.

The attribute name designates the role played by a domain in a
relation:
Used to interpret the meaning of the data elements corresponding
to that attribute
Example: The domain Date may be used to define two attributes
named Invoice-date and Payment-date with different meanings
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 10
Formal Definitions - State
The relation state is a subset of the Cartesian
product of the domains of its attributes
each domain contains the set of all possible values
the attribute can take.
Example: attribute Cust-name is defined over the
domain of character strings of maximum length
25
dom(Cust-name) is varchar(25)
The role these strings play in the CUSTOMER
relation is that of the name of a customer.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 11
Formal Definitions - Summary
Formally,
Given R(A1, A2, .........., An)
r(R) c dom (A1) X dom (A2) X ....X dom(An)
R(A1, A2, , An) is the schema of the relation
R is the name of the relation
A1, A2, , An are the attributes of the relation
r(R): a specific state (or "value" or population) of
relation R this is a set of tuples (rows)
r(R) = {t1, t2, , tn} where each ti is an n-tuple
ti = <v1, v2, , vn> where each vj element-of dom(Aj)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 12
Formal Definitions - Example
Let R(A1, A2) be a relation schema:
Let dom(A1) = {0,1}
Let dom(A2) = {a,b,c}
Then: dom(A1) X dom(A2) is all possible combinations:
{<0,a> , <0,b> , <0,c>, <1,a>, <1,b>, <1,c> }

The relation state r(R) c dom(A1) X dom(A2)
For example: r(R) could be {<0,a> , <0,b> , <1,c> }
this is one possible state (or population or extension) r of
the relation R, defined over A1 and A2.
It has three 2-tuples: <0,a> , <0,b> , <1,c>
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 13
Definition Summary
Informal Terms Formal Terms
Table Relation
All possible Column
Values
Domain
Row Tuple
Table Definition Schema of a Relation
Populated Table State of the Relation
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 14
Example A relation STUDENT
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 15
Characteristics Of Relations
Ordering of tuples in a relation r(R):
The tuples are not considered to be ordered,
even though they appear to be in the tabular
form.
Ordering of attributes in a relation schema R (and
of values within each tuple):
We will consider the attributes in R(A1, A2, ...,
An) and the values in t=<v1, v2, ..., vn> to be
ordered .
(However, a more general alternative definition of
relation does not require this ordering).
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 16
Same state as previous Figure (but
with different order of tuples)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 17
Characteristics Of Relations
Values in a tuple:
All values are considered atomic (indivisible).
Each value in a tuple must be from the domain of
the attribute for that column
If tuple t = <v1, v2, , vn> is a tuple (row) in the
relation state r of R(A1, A2, , An)
Then each vi must be a value from dom(Ai)

A special null value is used to represent values
that are unknown or inapplicable to certain tuples.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 18
Characteristics Of Relations
Notation:
We refer to component values of a tuple t by:
t[Ai] or t.Ai
This is the value vi of attribute Ai for tuple t
Similarly, t[Au, Av, ..., Aw] refers to the subtuple of
t containing the values of attributes Au, Av, ..., Aw,
respectively in t
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 19
Relational Algebra Overview
Relational algebra is the basic set of operations
for the relational model
These operations enable a user to specify basic
retrieval requests (or queries)
The result of an operation is a new relation, which
may have been formed from one or more input
relations
This property makes the algebra closed (all
objects in relational algebra are relations)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 20
Relational Algebra Overview (continued)
The algebra operations thus produce new
relations
These can be further manipulated using
operations of the same algebra
A sequence of relational algebra operations
forms a relational algebra expression
The result of a relational algebra expression is also a
relation that represents the result of a database
query (or retrieval request)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 21
Relational Algebra Overview
Relational Algebra consists of several groups of operations
Unary Relational Operations
SELECT (symbol: o (sigma))
PROJECT (symbol: t (pi))
RENAME (symbol: (rho))
Relational Algebra Operations From Set Theory
UNION ( ), INTERSECTION ( ), DIFFERENCE (or MINUS, )
CARTESIAN PRODUCT ( x )
Binary Relational Operations
JOIN (several variations of JOIN exist)
DIVISION
OUTER JOINS, OUTER UNION
AGGREGATE FUNCTIONS (These compute summary of
information: for example, SUM, COUNT, AVG, MIN, MAX)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 22
Unary Relational Operations: SELECT
The SELECT operation (denoted by o (sigma)) is used to select a
subset of the tuples from a relation based on a selection condition.
The selection condition acts as a filter
Keeps only those tuples that satisfy the qualifying condition
Tuples satisfying the condition are selected whereas the
other tuples are discarded (filtered out)
Examples:
Select the EMPLOYEE tuples whose department number is 4:
o
DNO = 4
(EMPLOYEE)
Select the employee tuples whose salary is greater than \$30,000:
o
SALARY > 30,000
(EMPLOYEE)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 23
Unary Relational Operations: SELECT
In general, the select operation is denoted by
o
<selection condition>
(R) where
the symbol o (sigma) is used to denote the select
operator
the selection condition is a Boolean (conditional)
expression specified on the attributes of relation R
tuples that make the condition true are selected
appear in the result of the operation
tuples that make the condition false are filtered out
discarded from the result of the operation
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 24
Unary Relational Operations: SELECT
(contd.)
SELECT Operation Properties
The SELECT operation o
<selection condition>
(R) produces a relation
S that has the same schema (same attributes) as R
SELECT o is commutative:
o
<condition1>
(o
< condition2>
(R)) = o
<condition2>
(o
< condition1>
(R))
Because of commutativity property, a cascade (sequence) of
SELECT operations may be applied in any order:
o
<cond1>
(o
<cond2>
(o
<cond3>
(R)) = o
<cond2>
(o
<cond3>
(o
<cond1>
( R)))
A cascade of SELECT operations may be replaced by a single
selection with a conjunction of all the conditions:
o
<cond1>
(o
< cond2>
(o
<cond3>
(R)) = o
<cond1> AND < cond2> AND <
cond3>
(R)))
The number of tuples in the result of a SELECT is less than
(or equal to) the number of tuples in the input relation R

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 25
The following query results refer to this
database state
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 26
Unary Relational Operations: PROJECT
PROJECT Operation is denoted by t (pi)
This operation keeps certain columns (attributes)
from a relation and discards the other columns.
PROJECT creates a vertical partitioning
The list of specified columns (attributes) is kept in
each tuple
The other attributes in each tuple are discarded
Example: To list each employees first and last
name and salary, the following is used:
t
LNAME, FNAME,SALARY
(EMPLOYEE)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 27
Unary Relational Operations: PROJECT
(cont.)

The general form of the project operation is:
t
<attribute list>
(R)
t (pi) is the symbol used to represent the project
operation
<attribute list> is the desired list of attributes from
relation R.
The project operation removes any duplicate
tuples
This is because the result of the project operation
must be a set of tuples
Mathematical sets do not allow duplicate elements.

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 28
Unary Relational Operations: PROJECT
(contd.)
PROJECT Operation Properties
The number of tuples in the result of projection
t
<list>
(R) is always less or equal to the number of
tuples in R
If the list of attributes includes a key of R, then the
number of tuples in the result of PROJECT is equal
to the number of tuples in R
PROJECT is not commutative
t
<list1>
(t
<list2>
(R) ) = t
<list1>
(R) as long as <list2>
contains the attributes in <list1>
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 29
Examples of applying SELECT and
PROJECT operations
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 30
Relational Algebra Expressions
We may want to apply several relational algebra
operations one after the other
Either we can write the operations as a single
relational algebra expression by nesting the
operations, or
We can apply one operation at a time and create
intermediate result relations.
In the latter case, we must give names to the
relations that hold the intermediate results.

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 31
Single expression versus sequence of
relational operations (Example)
To retrieve the first name, last name, and salary of all
employees who work in department number 5, we must
apply a select and a project operation
We can write a single relational algebra expression as
follows:
t
FNAME, LNAME, SALARY
(o
DNO=5
(EMPLOYEE))
OR We can explicitly show the sequence of operations,
giving a name to each intermediate relation:
DEP5_EMPS o
DNO=5
(EMPLOYEE)
RESULT t
FNAME, LNAME, SALARY
(DEP5_EMPS)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 32
Unary Relational Operations: RENAME
The RENAME operator is denoted by (rho)
In some cases, we may want to rename the
attributes of a relation or the relation name or
both
Useful when a query requires multiple
operations
Necessary in some cases (see JOIN operation
later)

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 33
Unary Relational Operations: RENAME
(contd.)
The general RENAME operation can be
expressed by any of the following forms:

S (B1, B2, , Bn )
(R) changes both:
the relation name to S, and
the column (attribute) names to B1, B1, ..Bn

S
(R) changes:
the relation name only to S

(B1, B2, , Bn )
(R) changes:
the column (attribute) names only to B1, B1, ..Bn
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 34
Unary Relational Operations: RENAME
(contd.)
For convenience, we also use a shorthand for
renaming attributes in an intermediate relation:
If we write:
RESULT t
FNAME, LNAME, SALARY
(DEP5_EMPS)
RESULT will have the same attribute names as
DEP5_EMPS (same attributes as EMPLOYEE)
If we write:
RESULT (F, M, L, S, B, A, SX, SAL, SU, DNO)

RESULT (F.M.L.S.B,A,SX,SAL,SU, DNO)
(DEP5_EMPS)
The 10 attributes of DEP5_EMPS are renamed to
F, M, L, S, B, A, SX, SAL, SU, DNO, respectively
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 35
Example of applying multiple operations
and RENAME
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 36
Relational Algebra Operations from
Set Theory: UNION
UNION Operation
Binary operation, denoted by
The result of R S, is a relation that includes all
tuples that are either in R or in S or in both R and
S
Duplicate tuples are eliminated
The two operand relations R and S must be type
compatible (or UNION compatible)
R and S must have same number of attributes
Each pair of corresponding attributes must be type
compatible (have same or compatible domains)

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 37
Relational Algebra Operations from
Set Theory: UNION
Example:
To retrieve the social security numbers of all employees who
either work in department 5 (RESULT1 below) or directly
supervise an employee who works in department 5 (RESULT2
below)
We can use the UNION operation as follows:
DEP5_EMPS o
DNO=5
(EMPLOYEE)
RESULT1 t
SSN
(DEP5_EMPS)
RESULT2(SSN) t
(DEP5_EMPS)
RESULT RESULT1 RESULT2
The union operation produces the tuples that are in either
RESULT1 or RESULT2 or both
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 38
Example of the result of a UNION
operation
UNION Example

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 39
Relational Algebra Operations from
Set Theory
Type Compatibility of operands is required for the binary
set operation UNION , (also for INTERSECTION , and
SET DIFFERENCE , see next slides)
R1(A1, A2, ..., An) and R2(B1, B2, ..., Bn) are type
compatible if:
they have the same number of attributes, and
the domains of corresponding attributes are type compatible
(i.e. dom(Ai)=dom(Bi) for i=1, 2, ..., n).
The resulting relation for R1R2 (also for R1R2, or R1
R2, see next slides) has the same attribute names as the
first operand relation R1 (by convention)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 40
Relational Algebra Operations from Set
Theory: INTERSECTION
INTERSECTION is denoted by
The result of the operation R S, is a
relation that includes all tuples that are in
both R and S
The attribute names in the result will be the
same as the attribute names in R
The two operand relations R and S must be
type compatible

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 41
Relational Algebra Operations from Set
Theory: SET DIFFERENCE (cont.)
SET DIFFERENCE (also called MINUS or
EXCEPT) is denoted by
The result of R S, is a relation that includes all
tuples that are in R but not in S
The attribute names in the result will be the
same as the attribute names in R
The two operand relations R and S must be
type compatible

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 42
Example to illustrate the result of UNION,
INTERSECT, and DIFFERENCE
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 43
Some properties of UNION, INTERSECT,
and DIFFERENCE
Notice that both union and intersection are commutative
operations; that is
R S = S R, and R S = S R
Both union and intersection can be treated as n-ary
operations applicable to any number of relations as both
are associative operations; that is
R (S T) = (R S) T
(R S) T = R (S T)
The minus operation is not commutative; that is, in
general
R S S R
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 44
Relational Algebra Operations from Set
Theory: CARTESIAN PRODUCT
CARTESIAN (or CROSS) PRODUCT Operation
This operation is used to combine tuples from two relations
in a combinatorial fashion.
Denoted by R(A1, A2, . . ., An) x S(B1, B2, . . ., Bm)
Result is a relation Q with degree n + m attributes:
Q(A1, A2, . . ., An, B1, B2, . . ., Bm), in that order.
The resulting relation state has one tuple for each
combination of tuplesone from R and one from S.
Hence, if R has n
R
tuples (denoted as |R| = n
R
), and S has
n
S
tuples, then R x S will have n
R
* n
S
tuples.
The two operands do NOT have to be "type compatible
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 45
Relational Algebra Operations from Set
Theory: CARTESIAN PRODUCT (cont.)
Generally, CROSS PRODUCT is not a
meaningful operation
Can become meaningful when followed by other
operations
Example (not meaningful):
FEMALE_EMPS o
SEX=F
(EMPLOYEE)
EMPNAMES t
FNAME, LNAME, SSN
(FEMALE_EMPS)
EMP_DEPENDENTS EMPNAMES x DEPENDENT
EMP_DEPENDENTS will contain every combination of
EMPNAMES and DEPENDENT
whether or not they are actually related

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 46
Relational Algebra Operations from Set
Theory: CARTESIAN PRODUCT (cont.)
To keep only combinations where the
DEPENDENT is related to the EMPLOYEE, we
add a SELECT operation as follows
Example (meaningful):
FEMALE_EMPS o
SEX=F
(EMPLOYEE)
EMPNAMES t
FNAME, LNAME, SSN
(FEMALE_EMPS)
EMP_DEPENDENTS EMPNAMES x DEPENDENT
ACTUAL_DEPS o
SSN=ESSN
(EMP_DEPENDENTS)
RESULT t
FNAME, LNAME, DEPENDENT_NAME
(ACTUAL_DEPS)
RESULT will now contain the name of female employees
and their dependents

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 47
Example of applying CARTESIAN
PRODUCT
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 48
Binary Relational Operations: JOIN
JOIN Operation (denoted by )
The sequence of CARTESIAN PRODECT followed by
SELECT is used quite commonly to identify and select
related tuples from two relations
A special operation, called JOIN combines this sequence
into a single operation
This operation is very important for any relational database
with more than a single relation, because it allows us
combine related tuples from various relations
The general form of a join operation on two relations R(A1,
A2, . . ., An) and S(B1, B2, . . ., Bm) is:
R
<join condition>
S
where R and S can be any relations that result from general
relational algebra expressions.

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 49
Binary Relational Operations: JOIN (cont.)
Example: Suppose that we want to retrieve the name of the
manager of each department.
To get the managers name, we need to combine each
DEPARTMENT tuple with the EMPLOYEE tuple whose SSN
value matches the MGRSSN value in the department tuple.
We do this by using the join operation.

DEPT_MGR DEPARTMENT
EMPLOYEE
Combines each department record with the employee who
manages the department
The join condition can also be specified as

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 50
Example of applying the JOIN operation
DEPT_MGR DEPARTMENT
EMPLOYEE

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 51
Some properties of JOIN
Consider the following JOIN operation:
R(A1, A2, . . ., An) S(B1, B2, . . ., Bm)
R.Ai=S.Bj
Result is a relation Q with degree n + m attributes:
Q(A1, A2, . . ., An, B1, B2, . . ., Bm), in that order.
The resulting relation state has one tuple for each
combination of tuplesr from R and s from S, but only if
they satisfy the join condition r[Ai]=s[Bj]
Hence, if R has n
R
tuples, and S has n
S
tuples, then the join
result will generally have less than n
R
* n
S
tuples.
Only related tuples (based on the join condition) will appear
in the result
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 52
Some properties of JOIN
The general case of JOIN operation is called a
Theta-join: R S
theta
The join condition is called theta
Theta can be any general boolean expression on
the attributes of R and S; for example:
R.Ai<S.Bj AND (R.Ak=S.Bl OR R.Ap<S.Bq)
Most join conditions involve one or more equality
conditions ANDed together; for example:
R.Ai=S.Bj AND R.Ak=S.Bl AND R.Ap=S.Bq

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 53
Binary Relational Operations: EQUIJOIN
EQUIJOIN Operation
The most common use of join involves join
conditions with equality comparisons only
Such a join, where the only comparison operator
used is =, is called an EQUIJOIN.
In the result of an EQUIJOIN we always have one
or more pairs of attributes (whose names need not
be identical) that have identical values in every
tuple.
The JOIN seen in the previous example was an
EQUIJOIN.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 54
Binary Relational Operations:
NATURAL JOIN Operation
NATURAL JOIN Operation
Another variation of JOIN called NATURAL JOIN
denoted by * was created to get rid of the second
(superfluous) attribute in an EQUIJOIN condition.
because one of each pair of attributes with identical values is
superfluous
The standard definition of natural join requires that the two
join attributes, or each pair of corresponding join attributes,
have the same name in both relations
If this is not the case, a renaming operation is applied first.

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 55
Binary Relational Operations
NATURAL JOIN (contd.)
Example: To apply a natural join on the DNUMBER attributes of
DEPARTMENT and DEPT_LOCATIONS, it is sufficient to write:
DEPT_LOCS DEPARTMENT * DEPT_LOCATIONS
Only attribute with the same name is DNUMBER
An implicit join condition is created based on this attribute:
DEPARTMENT.DNUMBER=DEPT_LOCATIONS.DNUMBER

Another example: Q R(A,B,C,D) * S(C,D,E)
The implicit join condition includes each pair of attributes with the
same name, ANDed together:
R.C=S.C AND R.D.S.D
Result keeps only one attribute of each such pair:
Q(A,B,C,D,E)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 56
Example of NATURAL JOIN operation
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 57
Complete Set of Relational Operations
The set of operations including SELECT o,
PROJECT t , UNION , DIFFERENCE ,
RENAME , and CARTESIAN PRODUCT X is
called a complete set because any other
relational algebra expression can be expressed
by a combination of these five operations.
For example:
R S = (R S ) ((R S) (S R))
R
<join condition>
S = o
<join condition>
(R X S)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 58
Binary Relational Operations: DIVISION
DIVISION Operation
The division operation is applied to two relations
R(Z) S(X), where X subset Z. Let Y = Z - X (and hence Z
= X Y); that is, let Y be the set of attributes of R that are
not attributes of S.

The result of DIVISION is a relation T(Y) that includes a
tuple t if tuples t
R
appear in R with t
R
[Y] = t, and with
t
R
[X] = t
s
for every tuple t
s
in S.

For a tuple t to appear in the result T of the DIVISION, the
values in t must appear in R in combination with every tuple
in S.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 59
Example of DIVISION
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 60
Recap of Relational Algebra Operations
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 61
Relational Calculus
A relational calculus expression creates a new
relation, which is specified in terms of variables
that range over rows of the stored database
relations (in tuple calculus) or over columns of
the stored relations (in domain calculus).
In a calculus expression, there is no order of
operations to specify how to retrieve the query
resulta calculus expression specifies only what
information the result should contain.
This is the main distinguishing feature between
relational algebra and relational calculus.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 62
Relational Calculus (Contd.)
Relational calculus is considered to be a
nonprocedural or declarative language.
This differs from relational algebra, where we
must write a sequence of operations to specify a
retrieval request; hence relational algebra can be
considered as a procedural way of stating a
query.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 63
Tuple Relational Calculus
The tuple relational calculus is based on specifying a
number of tuple variables.
Each tuple variable usually ranges over a particular
database relation, meaning that the variable may take as
its value any individual tuple from that relation.
A simple tuple relational calculus query is of the form
{t | COND(t)}
where t is a tuple variable and COND (t) is a conditional
expression involving t.
The result of such a query is the set of all tuples t that
satisfy COND (t).
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 64
Tuple Relational Calculus (Contd.)

Example: To find the first and last names of all employees
whose salary is above \$50,000, we can write the following
tuple calculus expression:
{t.FNAME, t.LNAME | EMPLOYEE(t) AND
t.SALARY>50000}
The condition EMPLOYEE(t) specifies that the range
relation of tuple variable t is EMPLOYEE.
The first and last name (PROJECTION t
FNAME, LNAME
) of
each EMPLOYEE tuple t that satisfies the condition
t.SALARY>50000 (SELECTION o
SALARY >50000
) will be
retrieved.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 65
The Existential and Universal Quantifiers
Two special symbols called quantifiers can appear in
formulas; these are the universal quantifier () and the
existential quantifier (-).
Informally, a tuple variable t is bound if it is quantified,
meaning that it appears in an ( t) or (- t) clause;
otherwise, it is free.
If F is a formula, then so are (- t)(F) and ( t)(F), where t
is a tuple variable.
The formula (- t)(F) is true if the formula F evaluates to true
for some (at least one) tuple assigned to free occurrences
of t in F; otherwise (- t)(F) is false.
The formula ( t)(F) is true if the formula F evaluates to
true for every tuple (in the universe) assigned to free
occurrences of t in F; otherwise ( t)(F) is false.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 66
The Existential and Universal
Quantifiers (Contd.)
is called the universal or for all quantifier
because every tuple in the universe of tuples
must make F true to make the quantified formula
true.
- is called the existential or there exists
quantifier because any tuple that exists in the
universe of tuples may make F true to make the
quantified formula true.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 67
Example Query Using Existential
Quantifier
Retrieve the name and address of all employees who work for the
Research department. The query can be expressed as :
{t.FNAME, t.LNAME, t.ADDRESS | EMPLOYEE(t) and (- d)
(DEPARTMENT(d) and d.DNAME=Research and
d.DNUMBER=t.DNO) }
The only free tuple variables in a relational calculus expression
should be those that appear to the left of the bar ( | ).
In above query, t is the only free variable; it is then bound
successively to each tuple.
If a tuple satisfies the conditions specified in the query, the attributes
FNAME, LNAME, and ADDRESS are retrieved for each such tuple.
The conditions EMPLOYEE (t) and DEPARTMENT(d) specify the
range relations for t and d.
The condition d.DNAME = Research is a selection condition and
corresponds to a SELECT operation in the relational algebra,
whereas the condition d.DNUMBER = t.DNO is a JOIN condition.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 68
Languages Based on Tuple Relational
Calculus
The language SQL is based on tuple calculus. It uses the
basic block structure to express the queries in tuple
calculus:
SELECT <list of attributes>
FROM <list of relations>
WHERE <conditions>
SELECT clause mentions the attributes being projected,
the FROM clause mentions the relations needed in the
query, and the WHERE clause mentions the selection as
well as the join conditions.
SQL syntax is expanded further to accommodate other
operations. (See Chapter 8).
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 69
Languages Based on Tuple Relational
Calculus (Contd.)
Another language which is based on tuple calculus is
QUEL which actually uses the range variables as in tuple
calculus. Its syntax includes:
RANGE OF <variable name> IS <relation name>
Then it uses
RETRIEVE <list of attributes from range variables>
WHERE <conditions>
This language was proposed in the relational DBMS
INGRES. (system is currently still supported by Computer
Associates but the QUEL language is no longer there).

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 70
The Domain Relational Calculus
Another variation of relational calculus called the domain relational
calculus, or simply, domain calculus is equivalent to tuple calculus
and to relational algebra.
The language called QBE (Query-By-Example) that is related to
domain calculus was developed almost concurrently to SQL at IBM
Research, Yorktown Heights, New York.
Domain calculus was thought of as a way to explain what QBE
does.
Domain calculus differs from tuple calculus in the type of variables
used in formulas:
Rather than having variables range over tuples, the variables
range over single values from domains of attributes.
To form a relation of degree n for a query result, we must have n of
these domain variables one for each attribute.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 71
The Domain Relational Calculus (Contd.)
An expression of the domain calculus is of the
form
{ x
1
, x
2
, . . ., x
n
|
COND(x
1
, x
2
, . . ., x
n
, x
n+1
, x
n+2
, . . ., x
n+m
)}
where x
1
, x
2
, . . ., x
n
, x
n+1
, x
n+2
, . . ., x
n+m
are
domain variables that range over domains (of
attributes)
and COND is a condition or formula of the domain
relational calculus.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 72
Example Query Using Domain Calculus
Retrieve the birthdate and address of the employee whose name is John B. Smith.
Query :
{uv | (- q) (- r) (- s) (- t) (- w) (- x) (- y) (- z)
(EMPLOYEE(qrstuvwxyz) and q=John and r=B and s=Smith)}

Abbreviated notation EMPLOYEE(qrstuvwxyz) uses the
variables without the separating commas: EMPLOYEE(q,r,s,t,u,v,w,x,y,z)

Ten variables for the employee relation are needed, one to range over the
domain of each attribute in order.
Of the ten variables q, r, s, . . ., z, only u and v are free.
Specify the requested attributes, BDATE and ADDRESS, by the free domain
variables u for BDATE and v for ADDRESS.
Specify the condition for selecting a tuple following the bar ( | )
namely, that the sequence of values assigned to the variables qrstuvwxyz be
a tuple of the employee relation and that the values for q (FNAME), r (MINIT),
and s (LNAME) be John, B, and Smith, respectively.
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 73
Retrieval Queries in SQL
SQL has one basic statement for retrieving information
from a database; the SELECT statement
This is not the same as the SELECT operation of the
relational algebra
Important distinction between SQL and the formal
relational model:
SQL allows a table (relation) to have two or more tuples that
are identical in all their attribute values
Hence, an SQL relation (table) is a multi-set (sometimes
called a bag) of tuples; it is not a set of tuples
SQL relations can be constrained to be sets by specifying
PRIMARY KEY or UNIQUE attributes, or by using the
DISTINCT option in a query
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 74
SET OPERATIONS
SQL has directly incorporated some set operations
There is a union operation (UNION), and in some
versions of SQL there are set difference (MINUS) and
intersection (INTERSECT) operations
The resulting relations of these set operations are sets of
tuples; duplicate tuples are eliminated from the result
The set operations apply only to union compatible
relations; the two relations must have the same attributes
and the attributes must appear in the same order
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 75
SET OPERATIONS (contd.)
Query 4: Make a list of all project numbers for projects that involve an
employee whose last name is 'Smith' as a worker or as a manager of
the department that controls the project.

Q4: (SELECT PNAME
FROM PROJECT, DEPARTMENT,
EMPLOYEE
WHERE DNUM=DNUMBER AND
UNION
(SELECT PNAME
FROM PROJECT, WORKS_ON, EMPLOYEE
WHERE PNUMBER=PNO AND
ESSN=SSN AND NAME='Smith')
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 76
NESTING OF QUERIES
A complete SELECT query, called a nested query, can be
specified within the WHERE-clause of another query,
called the outer query
Many of the previous queries can be specified in an
alternative form using nesting
Query 1: Retrieve the name and address of all employees
who work for the 'Research' department.

FROM EMPLOYEE
WHERE DNO IN (SELECT DNUMBER
FROM DEPARTMENT
WHERE DNAME='Research' )

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 77
NESTING OF QUERIES (contd.)
The nested query selects the number of the 'Research'
department
The outer query select an EMPLOYEE tuple if its DNO
value is in the result of either nested query
The comparison operator IN compares a value v with a
set (or multi-set) of values V, and evaluates to TRUE if v
is one of the elements in V
In general, we can have several levels of nested queries
A reference to an unqualified attribute refers to the
relation declared in the innermost nested query
In this example, the nested query is not correlated with
the outer query
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 78
CORRELATED NESTED QUERIES
If a condition in the WHERE-clause of a nested query references an
attribute of a relation declared in the outer query, the two queries are
said to be correlated
The result of a correlated nested query is different for each tuple
(or combination of tuples) of the relation(s) the outer query
Query 12: Retrieve the name of each employee who has a dependent
with the same first name as the employee.

Q12: SELECT E.FNAME, E.LNAME
FROM EMPLOYEE AS E
WHERE E.SSN IN
(SELECT ESSN
FROM DEPENDENT
WHERE ESSN=E.SSN AND
E.FNAME=DEPENDENT_NAME)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 79
CORRELATED NESTED QUERIES
(contd.)
In Q12, the nested query has a different result in the outer
query
A query written with nested SELECT... FROM...
WHERE... blocks and using the = or IN comparison
operators can always be expressed as a single block
query. For example, Q12 may be written as in Q12A

Q12A: SELECT E.FNAME, E.LNAME
FROM EMPLOYEE E, DEPENDENT D
WHERE E.SSN=D.ESSN AND
E.FNAME=D.DEPENDENT_NAME
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 80
THE EXISTS FUNCTION
EXISTS is used to check whether the result of a
correlated nested query is empty (contains no
tuples) or not
We can formulate Query 12 in an alternative form
that uses EXISTS as Q12B

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 81
THE EXISTS FUNCTION (contd.)
Query 12: Retrieve the name of each employee
who has a dependent with the same first name as
the employee.

Q12B: SELECT FNAME, LNAME
FROM EMPLOYEE
WHERE EXISTS (SELECT *
FROM DEPENDENT
WHERE SSN=ESSN
AND

FNAME=DEPENDENT_NAME)

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 82
THE EXISTS FUNCTION (contd.)
Query 6: Retrieve the names of employees who have no
dependents.

Q6: SELECT FNAME, LNAME
FROM EMPLOYEE
WHERE NOT EXISTS (SELECT *
FROM DEPENDENT
WHERE SSN=ESSN)
In Q6, the correlated nested query retrieves all
DEPENDENT tuples related to an EMPLOYEE tuple. If
none exist, the EMPLOYEE tuple is selected
EXISTS is necessary for the expressive power of SQL
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 83
EXPLICIT SETS
It is also possible to use an explicit
(enumerated) set of values in the WHERE-
clause rather than a nested query
Query 13: Retrieve the social security numbers of
all employees who work on project number 1, 2,
or 3.
Q13: SELECT DISTINCT ESSN
FROM WORKS_ON
WHERE PNO IN (1, 2, 3)
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 84
NULLS IN SQL QUERIES
SQL allows queries that check if a value is NULL (missing
or undefined or not applicable)
SQL uses IS or IS NOT to compare NULLs because it
considers each NULL value distinct from other NULL
values, so equality comparison is not appropriate.
Query 14: Retrieve the names of all employees who do
not have supervisors.
Q14: SELECT FNAME, LNAME
FROM EMPLOYEE
Note: If a join condition is specified, tuples with NULL values
for the join attributes are not included in the result
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 85
Joined Relations Feature
in SQL2
Can specify a "joined relation" in the FROM-
clause
Looks like any other relation but is the result of a
join
Allows the user to specify different types of joins
(regular "theta" JOIN, NATURAL JOIN, LEFT
OUTER JOIN, RIGHT OUTER JOIN, CROSS
JOIN, etc)

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 86
Joined Relations Feature
in SQL2 (contd.)
Examples:
Q8: SELECT E.FNAME, E.LNAME, S.FNAME, S.LNAME
FROM EMPLOYEE E S

can be written as:
Q8: SELECT E.FNAME, E.LNAME, S.FNAME, S.LNAME
FROM (EMPLOYEE E LEFT OUTER JOIN

Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 87
Joined Relations Feature
in SQL2 (contd.)
Examples:
FROM EMPLOYEE, DEPARTMENT
WHERE DNAME='Research' AND DNUMBER=DNO
could be written as:
FROM (EMPLOYEE JOIN DEPARTMENT
ON DNUMBER=DNO)
WHERE DNAME='Research
or as:
FROM (EMPLOYEE NATURAL JOIN
DEPARTMENT
AS DEPT(DNAME, DNO, MSSN, MSDATE)
WHERE DNAME='Research
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 88
Joined Relations Feature
in SQL2 (contd.)
Another Example: Q2 could be written as follows;
this illustrates multiple joins in the joined tables
Q2: SELECT PNUMBER, DNUM, LNAME,
FROM (PROJECT JOIN
DEPARTMENT ON
DNUM=DNUMBER) JOIN
EMPLOYEE ON
WHERE PLOCATION='Stafford
Copyright 2007 Ramez Elmasri and Shamkant B. Navathe Slide 3- 89
Summary of SQL Queries
A query in SQL can consist of up to six clauses,
but only the first two, SELECT and FROM, are
mandatory. The clauses are specified in the
following order:

SELECT <attribute list>
FROM <table list>
[WHERE <condition>]
[GROUP BY <grouping attribute(s)>]
[HAVING <group condition>]
[ORDER BY <attribute list>]
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