RELATIONAL ALGEBRA IN DATABASE MANAGEMENT SYSTEM

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

6.2
Database System Concepts - 6th
Edition
Outline
Outline
 Relational Algebra
 Tuple Relational Calculus
 Domain Relational Calculus

6.3
Edition
Relational Algebra
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.

6.4
Edition
Select Operation
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:
 dept_name=“Physics”(instructor)

6.5
Edition
Project Operation
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)
)
(
,
,
2
,
1
r
k
A
A
A 


6.6
Edition
Union Operation
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))

6.7
Edition
Set Difference Operation
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))

6.8
Edition
Set-Intersection Operation
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)

6.9
Edition
Cartesian-Product Operation
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.

6.10
Edition
Rename Operation
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 .
)
(
)
,...,
2
,
1
( E
n
A
A
A
x


6.11
Edition
Formal Definition
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

6.12
Edition
Tuple Relational Calculus

6.13
Edition
 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

6.14
Edition
Predicate Calculus Formula
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

6.15
Edition
Example Queries
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)}
{t | t  instructor  t [salary ]  80000}
Notice that a relation on schema (ID) is implicitly defined by
the query
Notice that a relation on schema (ID, name, dept_name, salary) is
implicitly defined by the query

6.16
Edition
Example Queries
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” ))}

6.17
Edition
Example Queries
Example Queries
 u  section (t [course_id ] = u [course_id ] 
 Find the set of all courses taught in the Fall 2009 semester, and in
the Spring 2010 semester
  u  section (t [course_id ] = u [course_id ] 
 Find the set of all courses taught in the Fall 2009 semester, but not in

6.18
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]))}

6.19
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.

6.20
Edition
Safety of Expressions (Cont.)
Safety of Expressions (Cont.)
 Consider again that query to 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]))}
 Without the existential quantification on student, the above
query would be unsafe if the Biology department has not
offered any courses.

6.21
Edition
Domain Relational Calculus

6.22
Edition
 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

6.23
Edition
Example Queries
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” ))}

6.24
Edition
Example Queries
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
( (s = “Fall”  y = 2009 ) v (s = “Spring”  y = 2010))}
 Find the set of all courses taught in the Fall 2009 semester, and in
s = “Fall”  y = 2009 )
  a, s, y, b, r, t ( <c, a, s, y, b, r, t >  section ] 
s = “Spring”  y = 2010)}

6.25
Edition
The expression:
{  x1, x2, …, xn  | P (x1, x2, …, xn )}
is safe if all of the following hold:
1. 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 ).
2. 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.
3. 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).

6.26
Edition
 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.

Database System Concepts, 6th
Ed.
See www.db-book.com for conditions on re-use
End of Chapter 6
End of Chapter 6

RELATIONAL ALGEBRA IN DATABASE MANAGEMENT SYSTEM

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