The CARTESIAN PRODUCT (CROSS PRODUCT) Operation
6.2.2 The CARTESIAN PRODUCT (CROSS PRODUCT) Operation
Next, we discuss the CARTESIAN PRODUCT operation—also known as CROSS PRODUCT or CROSS JOIN — which is denoted by ×. This is also a binary set opera- tion, but the relations on which it is applied do not have to be union compatible. In its binary form, this set operation produces a new element by combining every member (tuple) from one relation (set) with every member (tuple) from the other
relation (set). In general, the result of R(A 1 ,A 2 , ..., A n ) × S(B 1 ,B 2 , ..., B m ) is a rela- tion Q with degree n + m attributes Q(A 1 ,A 2 , ..., A n ,B 1 ,B 2 , ..., B m ), in that order. The resulting relation Q has one tuple for each combination of tuples—one 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 × S will have n R *n S tuples.
The n-ary CARTESIAN PRODUCT operation is an extension of the above concept, which produces new tuples by concatenating all possible combinations of tuples from n underlying relations.
In general, the CARTESIAN PRODUCT operation applied by itself is generally mean- ingless. It is mostly useful when followed by a selection that matches values of attributes coming from the component relations. For example, suppose that we want to retrieve a list of names of each female employee’s dependents. We can do this as follows:
FEMALE_EMPS ← σ Sex = ‘F’ ( EMPLOYEE ) EMPNAMES ← π Fname , Lname , Ssn ( FEMALE_EMPS ) EMP_DEPENDENTS ← EMPNAMES × DEPENDENT ACTUAL_DEPENDENTS ← σ Ssn = Essn ( EMP_DEPENDENTS )
RESULT ← π Fname , Lname , Dependent_name ( ACTUAL_DEPENDENTS ) The resulting relations from this sequence of operations are shown in Figure 6.5.
The EMP_DEPENDENTS relation is the result of applying the CARTESIAN PROD- UCT operation to EMPNAMES from Figure 6.5 with DEPENDENT from Figure 3.6. In EMP_DEPENDENTS , every tuple from EMPNAMES is combined with every tuple from DEPENDENT , giving a result that is not very meaningful (every dependent is combined with every female employee). We want to combine a female employee
156 Chapter 6 The Relational Algebra and Relational Calculus
Fname
FEMALE_EMPS Alicia
Jennifer Joyce
Zelaya Wallace
3321Castle, Spring, TX
987654321
Ssn
1972-07-31
1968-07-19 1941-06-20
Bdate
5631 Rice, Houston, TX
291Berry, Bellaire, TX
FF
Address
Sex Dno 25000
Super_ssn
Fname
EMPNAMES Alicia
Jennifer Joyce
English
Zelaya Wallace
EMP_DEPENDENTS Alicia
Alicia Alicia Alicia Alicia Alicia Alicia
Jennifer Jennifer Jennifer
Jennifer Jennifer Joyce
Jennifer
Jennifer
Joyce Joyce
Zelaya
Zelaya Zelaya
Zelaya
Zelaya Zelaya
Wallace
Wallace Wallace
Wallace
Wallace Wallace
Wallace English
Theodore Joy
FF
Dependent_name
1986-04-05 1958-05-03
1983-10-25
1942-02-28 1988-12-30
1988-01-04
1967-05-05 1983-10-25
1986-04-05
1958-05-03 1988-01-04
1942-02-28
1988-12-30 1986-04-05
1967-05-05 453453453
1983-10-25 1958-05-03
Bdate
Joyce Joyce
Joyce
Joyce English
English English
... 1988-01-04
1942-02-28 453453453
1988-12-30 1967-05-05
Fname
ACTUAL_DEPENDENTS Lname
Ssn
Essn
Dependent_name
Sex
Bdate ... Jennifer Wallace
987654321 Abner 987654321
1942-02-28 ...
Fname
RESULT Lname Dependent_name
Figure 6.5
The Cartesian Product (Cross Product) operation.
6.3 Binary Relational Operations: JOIN and DIVISION 157
Essn value match the Ssn value of the EMPLOYEE tuple. The ACTUAL_DEPENDENTS relation accomplishes this. The EMP_DEPENDENTS relation is a good example of the case where relational algebra can be correctly applied to yield results that make no sense at all. It is the responsibility of the user to make sure to apply only mean- ingful operations to relations.
The CARTESIAN PRODUCT creates tuples with the combined attributes of two rela- tions. We can SELECT related tuples only from the two relations by specifying an appropriate selection condition after the Cartesian product, as we did in the preced- ing example. Because this sequence of CARTESIAN PRODUCT followed by SELECT is quite commonly used to combine related tuples from two relations, a special oper-
ation, called JOIN , was created to specify this sequence as a single operation. We dis- cuss the JOIN operation next.
In SQL, CARTESIAN PRODUCT can be realized by using the CROSS JOIN option in joined tables (see Section 5.1.6). Alternatively, if there are two tables in the WHERE
clause and there is no corresponding join condition in the query, the result will also
be the CARTESIAN PRODUCT of the two tables (see Q10 in Section 4.3.3).
Parts
» Fundamentals_of_Database_Systems,_6th_Edition
» Characteristics of the Database Approach
» Advantages of Using the DBMS Approach
» A Brief History of Database Applications
» Schemas, Instances, and Database State
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» Discussion and Summary of SQL Queries
» Specifying General Constraints as Assertions in SQL
» Introduction to Triggers in SQL
» Specification of Views in SQL
» View Implementation, View Update, and Inline Views
» Schema Change Statements in SQL
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» The UNION, INTERSECTION, and MINUS Operations
» The CARTESIAN PRODUCT (CROSS PRODUCT) Operation
» Variations of JOIN: The EQUIJOIN and NATURAL JOIN
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» Examples of Queries in Relational Algebra
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» Implementing the SELECT Operation
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» Spatial Database Concepts 24
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