Division is typically required when you want to find out entities that are interacting with **all entities** of a set of different type entities.

The division operator is used when we have to evaluate queries which contain the keyword ‘all’.

**Some instances where division operator is used are:**

- Which person has account in all the banks of a particular city?
- Which students have taken all the courses required to graduate?

In all these queries, the description after the keyword ‘all’ defines a set which contains some elements and the final result contains those units who satisfy these requirements.

**Important: Division is not supported by SQL implementations. However, it can be represented using other operations.(like cross join, Except, In ) **

** SQL Implementation of Division **

** Given two relations(tables): R(x,y) , S(y). **

**R and S** : tables

**x and y** : column of R

**y** : column of S

** R(x,y) div S(y) ** means gives all distinct values of x from R that are associated with all values of y in S.

**Computation of Division :** R(x,y) div S(y)

**Steps:**

- Find out all possible combinations of S(y) with R(x) by computing R(x) x(cross join) S(y), say r1
- Subtract actual R(x,y) from r1, say r2
- x in r2 are those that are not associated with every value in S(y); therefore R(x)-r2(x) gives us x

that are associated with all values in S

**Queries**

**Implementation 1:****SELECT * FROM R WHERE x not in ( SELECT x FROM ( (SELECT x , y FROM (select y from S ) as p cross join (select distinct x from R) as sp) EXCEPT (SELECT x , y FROM R) ) AS r );****Implementation 2 :**Using correlated subquery**SELECT * FROM R as sx WHERE NOT EXISTS ( (SELECT p.y FROM S as p ) EXCEPT (SELECT sp.y FROM R as sp WHERE sp.x = sx.x ) );**

**Relational algebra**

Using steps which is mention above: All possible combinationsr1 ← πx(R) x Sx values with “incomplete combinations”,r2x ← πx(r1-R)andresult ← πx(R)-r2xR div S = πx(R)- πx((πx(R) x S) – R)

### Examples

**Supply Schema**

Here **sid** means **supplierID** and **pid** means **partsID**.

**Tables:** suppliers(sid,pid) , parts(pid)

** 1. Find suppliers that supply all parts. **

**Ans 1 : Using implementation 1 **

SELECT * FROM suppliers WHERE sid not in ( SELECT sid FROM ( (SELECT sid, pid FROM (select pid from parts) as p cross join (select distinct sid from supplies) as sp) EXCEPT (SELECT sid, pid FROM supplies)) AS r );

** Ans 2: Using implementation 2 **

SELECT * FROM suppliers as s WHERE NOT EXISTS (( SELECT p.pid FROM parts as p ) EXCEPT (SELECT sp.pid FROM supplies sp WHERE sp.sid = s.sid ) );

**Company schema**

**2. List employees who work on all projects controlled by dno=4. **

**Ans 1. Using implementation 1 **

SELECT * FROM employee AS e WHERE ssn NOT IN ( SELECT essn FROM ( (SELECT essn, pno FROM (select pno from project where dno=4) as p cross join (select distinct essn from works_on) as w) EXCEPT (SELECT essn, pno FROM works_on)) AS r );

**Ans 2. Using implementation 2**

SELECT * FROM employee AS e WHERE NOT EXISTS ( (SELECT pno FROM project WHERE dno = 4) EXCEPT (SELECT pno FROM works_on WHERE essn = e.ssn) );

**
Important :** For division correlated query seems simpler to write but may expensive to execute.

- List supplier who supply all ‘Red’ Parts.(supply schema)
- Retrieve the names of employees, who work on all the projects that ‘John Smith’ works (company schema)

**Some more Examples**.

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