The DIVISION Operation
6.3.4 The DIVISION Operation
The DIVISION operation, denoted by ÷, is useful for a special kind of query that sometimes occurs in database applications. An example is Retrieve the names of employees who work on all the projects that ‘John Smith’ works on. To express this query using the DIVISION operation, proceed as follows. First, retrieve the list of project numbers that ‘John Smith’ works on in the intermediate relation SMITH_PNOS :
SMITH ← σ Fname = ‘John’ AND Lname = ‘Smith’ ( EMPLOYEE ) SMITH_PNOS ← π Pno ( WORKS_ON
Essn = Ssn SMITH )
Next, create a relation that includes a tuple < Pno , Essn > whenever the employee whose Ssn is Essn works on the project whose number is Pno in the intermediate relation SSN_PNOS :
SSN_PNOS ← π Essn , Pno ( WORKS_ON ) Finally, apply the DIVISION operation to the two relations, which gives the desired
employees’ Social Security numbers:
SSNS ( Ssn ) ← SSN_PNOS ÷ SMITH_PNOS RESULT ← π Fname , Lname ( SSNS * EMPLOYEE )
The preceding operations are shown in Figure 6.8(a).
Figure 6.8
The DIVISION operation. (a) Dividing SSN_PNOS by SMITH_PNOS. (b) T ← R ÷ S.
(a)
(b)
SSN_PNOS
SMITH_PNOS
A B A 123456789
1 1 a1 b1 a1 123456789
2 2 a2 b1 a2 666884444
3 a3 b1 a3 453453453
1 a4 b1 453453453
a1 b2 T 333445555
2 SSNS
a3 b2 B 333445555
2 Ssn
a2 b3 b1 333445555
a3 b3 b4 333445555
20 a4 b3 999887777
30 a1 b4 999887777
10 a2 b4 987987987
10 a3 b4 987987987
6.3 Binary Relational Operations: JOIN and DIVISION 163
In general, the DIVISION operation is applied to two relations R(Z) ÷ S(X), where the attributes of R are a subset of the attributes of S; that is, X ⊆ Z. Let Y be the set of attributes of R that are not attributes of S; that is, Y = Z – X (and hence Z = X ∪ Y ). 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. This means that, 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. Note that in the formulation of the DIVISION operation, the tuples in the denominator relation S restrict the numerator relation R by selecting those tuples in the result that match all values present in the denomina- tor. It is not necessary to know what those values are as they can be computed by another operation, as illustrated in the SMITH_PNOS relation in the above example.
Figure 6.8(b) illustrates a DIVISION operation where X = {A}, Y = {B}, and Z = {A, B}. Notice that the tuples (values) b 1 and b 4 appear in R in combination with all three tuples in S; that is why they appear in the resulting relation T. All other values of B in R do not appear with all the tuples in S and are not selected: b 2 does not
appear with a 2 , and b 3 does not appear with a 1 .
The DIVISION operation can be expressed as a sequence of π, ×, and – operations as follows:
T1 ←π Y (R) T2 ←π Y ((S × T1) – R) T ← T1 – T2
The DIVISION operation is defined for convenience for dealing with queries that involve universal quantification (see Section 6.6.7) or the all condition. Most
RDBMS implementations with SQL as the primary query language do not directly implement division. SQL has a roundabout way of dealing with the type of query illustrated above (see Section 5.1.4, queries Q3A and Q3B ). Table 6.1 lists the various basic relational algebra operations we have discussed.
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