In relational algebra, the division operation is used to find all the tuples in one relation that match a specific pattern of tuples in another relation. The division operation is denoted by the symbol ÷, and it is defined as follows:
Given two relations R(A, B) and S(B), the result of the division operation R ÷ S is a new relation T(A-B) that contains all the values of A for which all the values of B in R are also in S.
The division operation can be expressed in terms of the basic relational algebra operations as follows:
R ÷ S = πA(R) - πA((πA(R) x S) - R)
This expression involves the use of projection (π), cross-product (x), and set-difference (-) operations.
A typical query that calls for division is one that involves finding all the employees who have worked on all the projects in a given list. For example, suppose we have two relations: Employee(EmpID, Name) and Project(ProjID, Name), and a third relation WorksOn(EmpID, ProjID) that lists the employees who work on each project. To find the employees who have worked on all the projects in a list, we can express the query as:
πName(Employee) ÷ πName(ProjectList)
Here, ProjectList is a relation that contains the list of projects we want to check. The result of this query will be a relation that contains only the names of the employees who have worked on all the projects in the list.
Unlike the join operation, the division operation is not given special treatment in database systems because it is rarely used in practice. The division operation is computationally expensive, and it requires a lot of resources to execute. In addition, many queries that require division can be expressed using simpler operations, such as joins and set operations, which are more efficient and easier to understand. As a result, most database systems do not provide a built-in division operator, and it is left to the user to express division as a combination of other operations.
