Relational Algebra Archana Gupta CS 157

What is Relational Algebra? Relational Algebra is formal description of how relational database operates. It is a procedural query language, i.e. user must define both “how” and “what” to retrieve. It consists of a set of operators that consume either one or two relations as input. An operator produces one relation as its output.

Introduction to Relational Algebra Introduced by E. F. Codd in Codd proposed such an algebra as a basis for database query languages.

Terminology Relation - a set of tuples. Tuple - a collection of attributes which describe some real world entity. Attribute - a real world role played by a named domain. Domain - a set of atomic values. Set - a mathematical definition for a collection of objects which contains no duplicates.

Algebra Operations Unary Operations - operate on one relation. These include select, project and rename operators. Binary Operations - operate on pairs of relations. These include union, set difference, division, cartesian product, equality join, natural join, join and semi-join operators.

Select Operator The Select operator selects tuples that satisfies a predicate; e.g. retrieve the employees whose salary is 30,000 б Salary = 30,000 (Employee) Conditions in Selection: Simple Condition: (attribute)(comparison)(attribute) (attribute)(comparison)(constant) Comparison: =,≠,≤,≥, Condition: combination of simple conditions with AND, OR, NOT

Select Operator Example NameAgeWeight Harry3480 Sally2864 George2970 Helena54 Peter3480 NameAgeWeight Harry3480 Helena54 Peter3480 Person б Age≥34 (Person) NameAgeWeight Helena54 б Age=Weight (Person)

Project Operator Project (∏) retrieves a column. Duplication is not permitted. e.g., name of employees: ∏ name (Employee) e.g., name of employees earning more than 80,000: ∏ name (б Salary>80,000 (Employee))

Project Operator Example NameAgeSalary Harry3480,000 Sally2890,000 George2970,000 Helena5454,280 Peter3440,000 Name Harry Sally George Helena Peter Employee ∏ name (Employee)

Project Operator Example NameAgeSalary Harry3480,000 Sally2890,000 George2970,000 Helena5454,280 Peter3440,000 Name Sally Employee б Salary>80,000 (Employee) NameAgeSalary Sally2890,000 ∏ name (б Salary>80,000 (Employee))

Cartesian Product In mathematics, it is a set of all pairs of elements (x, y) that can be constructed from given sets, X and Y, such that x belongs to X and y to Y. It defines a relation that is the concatenation of every tuple of relation R with every tuple of relation S.

Cartesian Product Example NameAgeWeight Harry3480 Sally2864 George2970 City San Jose Austin Person City Person X City NameAgeWeightCity Harry3480San Jose Sally2864San Jose George2970San Jose Harry3480Austin Sally2864Austin George2970Austin

Rename Operator In relational algebra, a rename is a unary operation written as ρ a / b (R) where: a and b are attribute names R is a relation The result is identical to R except that the b field in all tuples is renamed to an a field. Example, rename operator changes the name of its input table to its subscript, ρ employee (Emp) Changes the name of Emp table to employee

Rename Operator Example NameSalary Harry80,000 Sally90,000 George70,000 Helena54,280 Peter40,000 Employee ρ EmployeeName / Name (Employee) EmployeeNameSalary Harry80,000 Sally90,000 George70,000 Helena54,280 Peter40,000

Union Operator The union operation is denoted U as in set theory. It returns the union (set union) of two compatible relations. For a union operation r U s to be legal, we require that, r and s must have the same number of attributes. The domains of the corresponding attributes must be the same. As in all set operations, duplicates are eliminated.

Union Operator Example FNLN SusanYao RameshShah BarbaraJones AmyFord JimmyWang FNLN JohnSmith RicardoBrown SusanYao FrancisJohnson RameshShah Student Professor FNLN SusanYao RameshShah BarbaraJones AmyFord JimmyWang JohnSmith RicardoBrown FrancisJohnson Student U Professor

Intersection Operator Denoted as . For relations R and S, intersection is R  S. Defines a relation consisting of the set of all tuples that are in both R and S. R and S must be union-compatible. Expressed using basic operations: R  S = R – (R – S)

Intersection Operator Example FNLN SusanYao RameshShah BarbaraJones AmyFord JimmyWang FNLN JohnSmith RicardoBrown SusanYao FrancisJohnson RameshShah Student Professor FNLN SusanYao RameshShah Student  Professor

Set Difference Operator For relations R and S, Set difference R - S, defines a relation consisting of the tuples that are in relation R, but not in S. Set difference S – R, defines a relation consisting of the tuples that are in relation S, but not in R.

Set Difference Operator Example FNLN SusanYao RameshShah BarbaraJones AmyFord JimmyWang FNLN JohnSmith RicardoBrown SusanYao FrancisJohnson RameshShah Student Professor FNLN BarbaraJones AmyFord JimmyWang Student - Professor FNLN JohnSmith RicardoBrown FrancisJohnson Professor - Student

Division Operator The division operator takes as input two relations, called the dividend relation (r on scheme R) and the divisor relation (s on scheme S) such that all the attributes in S also appear in R and S is not empty. The output of the division operation is a relation on scheme R with all the attributes common with S.

Division Operator Example StudentTask FredDatabase1 FredDatabase2 FredCompiler1 EugeneDatabase1 SaraDatabase1 SaraDatabase2 EugeneCompiler1 Task Database1 Database2 Completed DBProject Student Fred Sara Completed / DBProject

Natural Join Operator Natural join is a dyadic operator that is written as R lXl S where R and S are relations. The result of the natural join is the set of all combinations of tuples in R and S that are equal on their common attribute names.

Natural Join Example NameEmpIDDeptName Harry3415Finance Sally2241Sales George3401Finance Harriet2202Sales DeptNameMgr FinanceGeorge SalesHarriet ProductionCharles Employee Dept NameEmpIDDeptNameMgr Harry3415FinanceGeorge Sally2241SalesHarriet George3401FinanceGeorge Harriet2202SalesHarriet Employee lXl Dept For an example, consider the tables Employee and Dept and their natural join:

Semijoin Operator The semijoin is joining similar to the natural join and written as R ⋉ S where R and S are relations. The result of the semijoin is only the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names.

Semijoin Example NameEmpIDDeptName Harry3415Finance Sally2241Sales George3401Finance Harriet2202Sales DeptNameMgr SalesHarriet ProductionCharles Employee Dept NameEmpIDDeptName Sally2241Sales Harriet2202Sales Employee ⋉ Dept For an example consider the tables Employee and Dept and their semi join:

Outerjoin Operator Left outer join The left outer join is written as R =X S where R and S are relations. The result of the left outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition to tuples in R that have no matching tuples in S. Right outer join The right outer join is written as R X= S where R and S are relations. The result of the right outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition to tuples in S that have no matching tuples in R.

Left Outerjoin Example NameEmpIDDeptName Harry3415Finance Sally2241Sales George3401Finance Harriet2202Sales DeptNameMgr SalesHarriet Employee Dept NameEmpIDDeptNameMgr Harry3415Financeω Sally2241SalesHarriet George3401Financeω Harriet2202SalesHarriet Employee =X Dept For an example consider the tables Employee and Dept and their left outer join:

Right Outerjoin Example NameEmpIDDeptName Harry3415Finance Sally2241Sales George3401Finance Harriet2202Sales DeptNameMgr SalesHarriet ProductionCharles Employee Dept NameEmpIDDeptNameMgr Sally2241SalesHarriet 2202SalesHarriet ωωProductionCharles Employee X= Dept For an example consider the tables Employee and Dept and their right outer join:

Full Outer join Example NameEmpIDDeptName Harry3415Finance Sally2241Sales George3401Finance Harriet2202Sales DeptNameMgr SalesHarriet ProductionCharles Employee Dept NameEmpIDDeptNameMgr Harry3415Financeω Sally2241SalesHarriet George3401Financeω Harriet2202SalesHarriet ωωProductionCharles Employee =X= Dept The outer join or full outer join in effect combines the results of the left and right outer joins. For an example consider the tables Employee and Dept and their full outer join:

References _algebra#Outer_join 57/cs157alecturenotes.htm Database System Concepts, 5 th edition, Silberschatz, Korth, Sudarshan