Consider the following SQL to find which departments have had employees on the `Further Accounting' course.
SELECT DISTINCT dname FROM department, course, empcourse, employee WHERE cname = `Further Accounting' AND course.courseno = empcourse.courseno AND empcourse.empno = employee.empno AND employee.depno = department.depno;
The equivalent relational algebra is
PROJECTdname (department JOINdepno = depno ( PROJECTdepno (employee JOINempno = empno ( PROJECTempno (empcourse JOINcourseno = courseno ( PROJECTcourseno (SELECTcname = `Further Accounting' course) )) )) ))
From the example, one can see that for complicated cases a large amount of the answer is formed from operator names, such as PROJECT and JOIN. It is therefore commonplace to use symbolic notation to represent the operators.
The symbolic operators are used as with the verbal ones. So, to find all employees in department 1:
SELECTdepno = 1(employee) becomes σdepno = 1(employee)
Conditions can be combined together using ^ (AND) and v (OR). For example, all employees in department 1 called `Smith':
SELECTdepno = 1 ^ surname = `Smith'(employee) becomes σdepno = 1 ^ surname = `Smith'(employee)
The use of the symbolic notation can lend itself to brevity. Even better, when the JOIN is a natural join, the JOIN condition may be omitted from |x|. The earlier example resulted in:
PROJECTdname (department JOINdepno = depno ( PROJECTdepno (employee JOINempno = empno ( PROJECTempno (empcourse JOINcourseno = courseno ( PROJECTcourseno (SELECTcname = `Further Accounting' course)))))))
πdname (department |×| ( πdepno (employee |×| ( πempno (empcourse |×| ( πcourseno (σcname = `Further Accounting' course) ))))))
The rename operator returns an existing relation under a new name. ρA(B) is the relation B with its name changed to A. For example, find the employees in the same Department as employee 3.
ρemp2.surname,emp2.forenames ( σemployee.empno = 3 ^ employee.depno = emp2.depno ( employee × (ρemp2employee) ) )
A|×|cB ⇔ πa1,a2,...aN(σc(A × B))
c is called the join-condition, and is usually the comparison of primary and foreign key. Where there are N tables, there are usually N-1 join-conditions. In the case of a natural join, the conditions can be missed out, but otherwise missing out conditions results in a cartesian product (a common mistake to make).
The same relational algebraic expression can be written in many different ways. The order in which tuples appear in relations is never significant.
While equivalent expressions always give the same result, some may be much easier to evaluate that others.
When any query is submitted to the DBMS, its query optimiser tries to find the most efficient equivalent expression before evaluating it.
Any relational language as powerful as relational algebra is called relationally complete.
A relationally complete language can perform all basic, meaningful operations on relations.
Since SQL is a superset of relational algebra, it is also relationally complete.