DIVISION Operation

 

DIVISION Operation

Most relational algebra operations (SELECT, PROJECT, JOIN) answer existential queries:

  • Does there exist a matching tuple?

  • Find employees who work on project X

But some queries are universal in nature:

Find employees who work on all projects that John Smith works on

This “for all” requirement is exactly what the DIVISION (÷) operation is designed for.

👉 DIVISION captures universal quantification (“all”, “every”).

Intuition Behind DIVISION

Think of DIVISION as answering:

Which values are associated with every value in another set?

In general terms:

  • We have a numerator relation (larger relation)

  • We divide it by a denominator relation (a set of required values)

  • The result contains those tuples that are related to all denominator values

Example 

Query:

Retrieve the names of employees who work on all the projects that ‘John Smith’ works on

Step-by-Step Explanation

Step 1: Find John Smith

SMITH ← σFname='John' AND Lname='Smith'(EMPLOYEE)

  • This selects the EMPLOYEE tuple for John Smith.

  • We now know Smith’s Ssn.

Step 2: Find Projects John Smith Works On

SMITH_PNOS ← πPno(WORKS_ON ⨝Essn=Ssn SMITH)

  • Join WORKS_ON with SMITH

  • Project only the project numbers

  • Result:
    SMITH_PNOS(Pno) = all projects that John Smith works on

This relation becomes the denominator in the division.

Step 3: Create Employee–Project Pairs

SSN_PNOS ← πEssn, Pno(WORKS_ON)

  • This relation lists every employee and every project they work on

  • Schema:
    SSN_PNOS(Essn, Pno)

This is the numerator in the division.

Step 4: Apply DIVISION

SSNS(Ssn) ← SSN_PNOS ÷ SMITH_PNOS

Meaning:

Select those Essn values such that for every project in SMITH_PNOS,
the pair ⟨Essn, Pno⟩ exists in SSN_PNOS

In plain English:

👉 Employees who work on all projects that John Smith works on

The result contains only Ssn values.

Step 5: Get Employee Names

RESULT ← πFname, Lname(SSNS ⨝ EMPLOYEE)

  • Join the resulting SSNs with EMPLOYEE

  • Project names

  • Final answer: employee names


Formal Definition of DIVISION 

Let:

  • R(Z) be the numerator relation

  • S(X) be the denominator relation

  • X ⊆ Z

  • Y = Z − X

Then:

The result of R ÷ S is a relation T(Y) such that a tuple t is in T
if and only if, for every tuple tS in S, there exists a tuple tR in R where:

  • tR[X] = tS

  • tR[Y] = t

📌 This is the key sentence students must understand.

Meaning of the DIVISION Operation

For a tuple t to appear in the result T of a DIVISION operation:

  • The values in t must appear in the numerator relation R

  • In combination with every tuple in the denominator relation S

In other words:

A tuple is included in the result only if it is associated with all values present in the denominator relation.

Role of the Denominator Relation

  • The denominator relation S acts as a restricting relation

  • It does not contribute attributes to the result

  • Instead, it specifies a set of required values

The DIVISION operation selects only those tuples from the numerator relation R that:

  • Match every value in S

  • Satisfy the condition of universal participation

Key Insight

  • The numerator relation R provides the candidate tuples

  • The denominator relation S enforces the “for all” condition

  • Only those tuples in R that are related to all tuples in S appear in the result

Visual Interpretation 

What Happens Conceptually?

  • R contains many (A, B) pairs

  • S contains required A values

  • DIVISION returns those B values that appear with every A in S

In Figure 8.8(b):

  • b1 and b4 appear with all A values in S → included

  • b2, b3 miss at least one A → excluded




Why DIVISION Is Hard to Implement Directly

  • DIVISION is not primitive

  • It can be expressed using PROJECT, CARTESIAN PRODUCT, and SET DIFFERENCE

From the book:

T1 ← πY(R)
T2 ← πY((S × T1) – R)
T  ← T1 – T2

This formulation:

  • Generates candidates

  • Eliminates those missing required combinations

Key Points

✔ DIVISION answers “for all” queries
✔ Denominator specifies required values
✔ Result includes only tuples satisfying every requirement
✔ SQL does not directly support DIVISION
✔ Implemented using NOT EXISTS, GROUP BY + HAVING, or nested queries in SQL

 Summary 

DIVISION returns those tuples that are related to all tuples in another relation.

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