Relational Algebra Chapter 4 CIS 458 Sungchul Hong

Last Class Relational data model. Transform ERD to relational database schema. –One-to-one relationship –One-to-many relationship –Many-to-many relationship

Today Relational Algebra –Theoretical language –Similar to SQL –Data Retrieval concept

Sets A set is any well-defined list, collection, or class of objects. Elements or members of the set. A = {1, 3, 7, 10} B= {x | x is even}

Subsets If every element in a set A is also a member of a set B, then A is called a subset of B. A  B (Proper)

Union The union of sets A and B is the set of all elements which belong to A or to B or to both. A  B S={a,b,c,d} T={f,b,d,g}

The Relational Algebra The relational algebra is a theoretical language with operations that work on one or more relations to define another relation without changing the original relation(s). Operands and results are relations. –  name, address (Student) The output from one operation can become the input to another operation. –  name=“John” (  name, address (Student))

Five Basic Operations Selection:  Projection:  Cartesian product: X Union:  –Intersection:  Set difference: –

Selection  salary>10000 (Staff) AND, OR, NOT , , ~  salary>10000 (Staff)

Projection  staffNo,fName, lName, salary (Staff)

Set Operations

Union R  S  city (Branch)   city (PropertyForRent)

City London Aberdeen Glasgow Bristol  city (Branch)   city (PropertyForRent) Result table

Set Difference R – S  city (Branch) -  city (PropertyForRent)

city Bristol  city (Branch) -  city (PropertyForRent) Result

Intersection  city (Branch)   city (PropertyForRent) R  S = R – (R –S)

 city (Branch)   city (PropertyForRent)

city Aberdeen London Glasgow  city (Branch)   city (PropertyForRent) Result

Division R  S The Division operation defines a relation over the attributes C that consists of the set of tuples from R that match the combination of every tuple in S.

List the elements of A in V that satisfies all the condition in B (W).

NameCondition JohnSmart JohnKind TomSmart TomKind NickSmart Condition Smart Kind V W V  W = Name John Tom Match Making

Cartesian Product  clinetNo,fName,lName (Client) X (  clientNo,propertyNo,comment (Viewing))  clinet.clinetNo=Viewing.clientNo (  clinetNo,fName,lName (Client) X (  clientNo,propertyNo,comment (Viewing)))

 clinetNo,fName,lName (Client) X (  clientNo,propertyNo,comment (Viewing))

 clinet.clinetNo=Viewing.clientNo (  clinetNo,fName,lName (Client) X (  clientNo,propertyNo,comment (Viewing)))

Application Join –Selection (  ) and Cartesian product (X) One of the major database data retrieval technique.

Join Operations Theta join Equijoin Natural join –Outer join –Semijoin

Theta Join R ⊳⊲ F S The Theta join operation defines a relation that contains tuples satisfying the predicate F from the Cartesian product of R and S. The predicate F is of the form R.a j  S.b i where  may be one of the comparison operators (, , =,  ) –E.g.) R ⊳⊲ R.salary  S.salary S

Equijoin  clinetNo,fName,lName (Client) ⊳⊲ clinet.clinetNo=Viewing.clinetNo  clientNo,propertyNo,comment (Viewing) A special case of theta join.

Natural Join R ⊳⊲ S The Natural join is an Equijoin of the two relations R and S over all common attributes x. One occurrence of each common attribute is eliminated from the result.  clinetNo,fName,lName (Client) ⊳⊲  clientNo,propertyNo,comment (Viewing) –No common attribute?

Other Topics Outer Join Aggregation (COUNT, SUM AVG, MIN, MAX), Group Relational Calculus –Tuple Relational Calculus {S | Staff(S)  S.salary > 10000} –Domain Relational Calculus List the staff who manage properties for rent in Glasgow {sN, fN, lN, posn, sex, DOB, sal, bN|(  sN1, city)(Staff(sN, fN, lN, posn, sex, DOB, sal, bN)  PropertyForRent(pN, st, cty, pc, typ, rms, rnt, oN, sN1, bN1)  (sN = sN1)  cty=‘Glasgow’}