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Chapter 7
Relational Algebra
Prof. Yin-Fu Huang
CSIE, NYUST
7.1 Introduction
 Eight operators of the relational algebra:
1. The traditional set operators union, intersection, difference,
and Cartesian product.
2. The special relational operators restrict, project, join, and
divide. (See Fig. 7.1)
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7.2 Closure Revisited
 The output from any given relational operation is another relation.
 Nested relational expressions
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7.3 The Original Algebra: Syntax
 (See Page178-179)
<relation exp> ::= Relation {<tuple exp commalist>}
| <relvar name>
| <relation op inv>
| <with exp>
| <introduced name>
| (<relation exp>)
<relation op inv> ::= <project> | <nonproject>
<project> ::= <relation exp> { [All But] <attribute name commalist> }
<nonproject> ::= <rename> | <union> | <intersect> | <minus>
| <times> | <where> | <join> | <divide>
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7.3 The Original Algebra: Syntax (Cont.)
<rename> ::= <relation exp> Rename ( <renaming commalist> )
<union> ::= <relation exp> Union <relation exp>
<intersect> ::= <relation exp> Intersect <relation exp>
<minus> ::= <relation exp> Minus <relation exp>
<times> ::= <relation exp> Times <relation exp>
<where> ::= <relation exp> Where <bool exp>
<join> ::= <relation exp> Join <relation exp>
<divide> ::= <relation exp> Divideby <relation exp> Per <per>
<per> ::= <relation exp> | ( <relation exp>, <relation exp> )
<with exp> ::= With <name intro commalist> : <exp>
<name intro> ::= <exp> As <introduced name>
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7.4
The Original Algebra: Semantics
 Tuple-homogeneous
 Union, Intersect, and Difference (See Fig. 7.2)
 Product (See Fig. 7.3)
If we need to construct the Cartesian product of two relations
that do have any such common attribute names, we must use
the Rename operator first to rename attributes appropriately.
 Restrict (See Fig. 7.4)
 Project (See Fig. 7.5)
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Fig. 7.2
Union, intersection, and difference
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Fig. 7.3
Cartesian product example
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Fig. 7.4
Restriction examples
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Fig. 7.5
Projection examples
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7.4
The Original Algebra: Semantics (Cont.)
 Join
• natural join (See Fig. 7.6)
• θ-join (See Fig. 7.7)
((S Rename City As Scity)
Times
(P Rename City As Pcity))
Where Scity > Pcity
 Divide (See Fig. 7.8)
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Fig. 7.6
&
Fig. 7.7
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Fig. 7.8
Division examples
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7.5 Examples
 Exam 1: ((Sp Join S) Where P#=P#(‘P2’)) {Sname}
 Exam 2: (((P Where Color=Color(‘Red’))
Join Sp ) {S#} Join S) {Sname}
 Exam 3: ((S {S#} Divideby P {P#} Per Sp {S#, P#})
Join S) {Sname}
 Exam 4: S {S#} Divideby (Sp Where S#=S#(‘S2’)) {P#}
Per Sp {S#, P#}
 Exam 5: (((S Rename S# As Sa) {Sa, City} Join
(S Rename S# As Sb) {Sb, City})
Where Sa < Sb) {Sa, Sb}
 Exam 6: ((S {S#} Minus (Sp Where P#=P#(‘P2’)) {S#})
Join
S) {Sname}
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7.6 What Is the Algebra For?
 The operators join, intersect, and divide can be defined in
terms of the other five.
 Of the remaining five, however, none can be defined in terms of
the other four, so we can regard those five as constituting a
primitive or minimum set.
 Some possible applications:
1. Defining a scope for retrieval
2. Defining a scope for update
3. Defining integrity constraints
4. Defining derived relvars
5. Defining stability requirements
6. Defining security constraints
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7.6 What Is the Algebra For? (Cont.)
 A high-level, symbolic representation of the user‘s intent
 Transformation rules
((Sp Join S) Where P#=P#(‘P2’)) {Sname}
((Sp Where P#=P#(‘P2’)) Join S) {Sname}
 The algebra thus serves as a convenient basis for optimization.
 A language is said to be relationally complete if it is at least as
powerful as the algebra.
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7.7 Further Points
 Associativity and Commutativity
• Associative: Union, Intersect, Times, Join
e.g. (A Union B) Union C = A Union (B Union C)
= A Union B Union C
• Commutative: Union, Intersect, Times, Join
e.g. A Union B = B Union A
 Some Equivalences
e.g. r { } = Table_Dum if r=empty,
Table_Dee otherwise (a nullary projection)
r Join Table_Dee = Table_Dee Join r = r
r Times Table_Dee = Table_Dee Times r = r
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7.7 Further Points (Cont.)
 Some Generalizations
If s contains no relations at all, then:
The join of all relations in s is defined to be Table_Dee.
The union of all relations in s is defined to be the empty
relation.
The intersection of all relations in s is defined to be the
“universal” relation.
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7.8 Additional Operators
 (See Page 196)
<semijoin> ::= <relation exp> Semijoin <relation exp>
<semiminus> ::= <relation exp> Semiminus <relation exp>
<extend> ::= Extend <relation exp> Add ( <extend add commalist> )
<extend add> ::= <exp> As <attribute name>
<summarize> ::= Summarize <relation exp> Per <relation exp>
Add ( <summarize add commalist> )
<summarize add> ::= <summary type> [ ( <scalar type> ) ]
As <attribute name>
<summary type> ::= Count | Sum | Avg | Max | Min | All | Any
| Countd | Sumd | Avgd| …
<tclose> ::= Tclose <relation exp>
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7.8 Additional Operators (Cont.)
 Semijoin  (a Join b) {X, Y}
e.g. S Semijoin (Sp Where P#=P#(‘P2’))
 Semidifference  a Minus (a Semijoin b)
e.g. S Semiminus (Sp Where P#=P#(‘P2’))
 Extend
e.g. Extend P Add (Weight＊454) As Gmwt
(See Fig. 7.9)
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7.8 Additional Operators (Cont.)
 Exam 1: Extend S Add ‘Supplier’ As Tag
 Exam 2: Extend (P Join Sp) Add (Weight＊Qty) As Shipwt
 Exam 3: (Extend S Add City As Scity) {All But City}
Rename
 Exam 4: Extend P Add (Weight＊454 As Gmwt,
Weight＊16 As Ozwt)
 Exam 5: Extend S
Add Count((Sp Rename S# As X) Where X=S#)
As Np
(See Fig. 7.10)
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7.8 Additional Operators (Cont.)
 Summarize
e.g. Summarize Sp Per P {P#} Add Sum(Qty) As Totqty
(See Fig. 7.11)
 Exam 1: Summarize Sp Per P {P#} Add (Sum(Qty) As Totqty,
Avg(Qty) As Avgqty)
 Exam 2: Summarize Sp Per S {S#} Add Count As Np
• Summarize is not a primitive operator.
Extend
 Exam 3: Summarize S Per S {City}Add Avg(Status) As Avg_Status
 Exam 4: Summarize Sp Per Sp { }Add Sum(Qty) As Grandtotal
 Tclose  the transitive closure of a
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7.9 Grouping and Ungrouping
 Group
e.g. SP Group (P#, Qty) As PQ (See Fig. 7.12)
 Ungroup
e.g. SPQ Ungroup PQ
 The reversibility of the Group and Ungroup
operations
(See Fig. 7.13)
 Functionally dependency
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The End.
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