Introduction

The Division Operation is a binary operation in relational algebra that answers “for all” type queries. It finds tuples in one relation that are associated with all tuples in another relation.

Syntax

R ÷ S

Here, R is the dividend relation with attributes A and B, and S is the divisor relation with attribute A. The result contains only attribute B.

Example

Consider the following relations:

Relation R(A, B):
A  | B
A1 | B1
A2 | B1
A1 | B2
A2 | B2
A1 | B3

Relation S(A):
A
A1
A2
            

Query: Find all B values in R that are associated with all A values in S.

Result (R ÷ S):
B
B1
B2
            

Explanation: B1 and B2 are associated with both A1 and A2, while B3 is not associated with A2.

How Division Works

1. Project: Identify all distinct B values in R.
2. Cartesian Product: Combine S with the projected B values.
3. Set Difference: Remove tuples of R from the Cartesian Product to find missing pairs.
4. Final Result: Exclude B values with missing pairs to get the result.

Deriving Division Using Basic Operations

The Division Operation can be expressed using basic operations:

1. Step 1: Project distinct B values from R.
2. Step 2: Compute the Cartesian Product of S and the projected B values.
3. Step 3: Subtract R from the Cartesian Product to find missing combinations.
4. Step 4: Exclude missing B values to get the final result.

Use Cases of Division Operation

• Finding employees who have completed all training courses.
• Identifying students enrolled in all mandatory subjects.
• Listing products available in all stores.

Conclusion

• The Division Operation is crucial for answering “for all” queries in relational algebra.
• It simplifies queries where one attribute must be related to all instances of another attribute.
• Using basic operations like projection, Cartesian product, and set difference, we can derive the division operation.