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# Relational algebra-and-relational-calculus

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Relational algebra-and-relational-calculus presentation. its help in the understanding of DBMS course.

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### Relational algebra-and-relational-calculus

1. 1. Chapter 4 Relational Algebra and Relational Calculus Transparencies
2. 2. Relational Algebra Relational algebra and relational calculus are formal languages associated with the relational model. The Relational Algebra is used to define the ways in which relations (tables) can be operated to manipulate their data. It is used as the basis of SQL for relational databases, and illustrates the basic operations required of any DML. 2
3. 3. 3 Relational Algebra Relational algebra operations work on one or more relations to define another relation without changing the original relations. Both operands and results are relations, so output from one operation can become input to another operation. Allows expressions to be nested, just as in arithmetic. This property is called closure.
4. 4. 4 Relational algebra VS Relational Calculus Informally, relational algebra is a (high-level) procedural language and relational calculus a non-procedural language. – Difference ?? However, formally both are equivalent to one another. A language that produces a relation that can be derived using relational calculus is relationally complete. What & How
5. 5. 5 Relational Algebra This Algebra is composed of Unary operations (involving a single table) and Binary operations (involving multiple tables). Five basic operations in relational algebra: Selection, Projection, Cartesian product, Union, and Set Difference. These perform most of the data retrieval operations needed. Also have Join, Intersection, and Division operations, which can be expressed in terms of 5 basic operations.
6. 6. 6 Relational Algebra Operations
7. 7. 7 Relational Algebra Operations
8. 8. 8 Selection (or Restriction) σpredicate (R) – Works on a single relation R and defines a relation that contains only those tuples (rows) of R that satisfy the specified condition (predicate). – Unary Operation σ <condition> < tablename > Conditions in Selection: Simple Condition: (attribute)(comparison)(attribute) (attribute)(comparison)(constant) Comparison: =,≠,≤,≥,<,>
9. 9. Select Operator Example Name Age Weight Harry 34 80 Sally 28 64 George 29 70 Helena 54 54 Peter 34 80 Name Age Weight Harry 34 80 Helena 54 54 Peter 34 80 Person бAge≥34(Person) Name Age Weight Helena 54 54 бAge=Weight(Person)
10. 10. 10 Example - Selection (or Restriction) List all staff with a salary greater than £10,000. σsalary >10000 (Staff)
11. 11. 11 Projection Πcol1, .. . ,coln(R) – Works on a single relation R and defines a relation that contains a vertical subset of R, extracting the values of specified attributes and eliminating duplicates. π <columnlist > < tablename > e.g., name of employees: ∏ name(Employee) e.g., name of employees earning more than 80,000: ∏ name(бSalary>80,000(Employee))
12. 12. Project Operator Example Name Age Salary Harry 34 80,000 Sally 28 90,000 George 29 70,000 Helena 54 54,280 Peter 34 40,000 Name Harry Sally George Helena Peter Employee ∏ name(Employee)
13. 13. Project Operator Example Name Age Salary Harry 34 80,000 Sally 28 90,000 George 29 70,000 Helena 54 54,280 Peter 34 40,000 Name Sally Employee бSalary>80,000(Employee) Name Age Salary Sally 28 90,000 ∏ name(бSalary>80,000(Employee))
14. 14. 14 Example - Projection Produce a list of salaries for all staff, showing only staffNo, fName, lName, and salary details. ΠstaffNo, fName,lName, salary(Staff)
15. 15. Union, Intersection, Set-Difference All of these operations take two input relations, which must be union-compatible: – Same number of fields. – `Corresponding’ fields have the same type. 15
16. 16. 16 Union R ∪ S – Union of two relations R and S defines a relation that contains all the tuples of R, or S, or both R and S, duplicate tuples being eliminated. – R and S must be union-compatible. If R and S have I and J tuples, respectively, union is obtained by concatenating them into one relation with a maximum of (I + J) tuples.
17. 17. Union Operator Example FN LN Susan Yao Ramesh Shah Barbara Jones Amy Ford Jimmy Wang FN LN John Smith Ricardo Brown Susan Yao Francis Johnson Ramesh Shah Student Professor FN LN Susan Yao Ramesh Shah Barbara Jones Amy Ford Jimmy Wang John Smith Ricardo Brown Francis Johnson Student U Professor
18. 18. 18 Example - Union List all cities where there is either a branch office or a property for rent. Πcity(Branch) ∪ Πcity(PropertyForRent)
19. 19. 19 Set Difference R – S – Defines a relation consisting of the tuples that are in relation R, but not in S. – R and S must be union-compatible.
20. 20. Set Difference Operator Example FN LN Susan Yao Ramesh Shah Barbara Jones Amy Ford Jimmy Wang FN LN John Smith Ricardo Brown Susan Yao Francis Johnson Ramesh Shah Student Professor FN LN Barbara Jones Amy Ford Jimmy Wang Student - Professor FN LN John Smith Ricardo Brown Francis Johnson Professor - Student
21. 21. 21 Example - Set Difference List all cities where there is a branch office but no properties for rent. Πcity(Branch) – Πcity(PropertyForRent)
22. 22. 22 Intersection 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)
23. 23. Intersection Operator Example FN LN Susan Yao Ramesh Shah Barbara Jones Amy Ford Jimmy Wang FN LN John Smith Ricardo Brown Susan Yao Francis Johnson Ramesh Shah Student Professor FN LN Susan Yao Ramesh Shah Student ∩ Professor
24. 24. 24 Example - Intersection List all cities where there is both a branch office and at least one property for rent. Πcity(Branch) ∩ Πcity(PropertyForRent)
25. 25. 25 Cartesian product R X S – Defines a relation that is the concatenation of every tuple of relation R with every tuple of relation S.
26. 26. 26 Example - Cartesian product List the names and comments of all clients who have viewed a property for rent. (ΠclientNo,fName,lName(Client)) X (ΠclientNo,propertyNo,comment(Viewing))
27. 27. 27 Example - Cartesian product and Selection Use selection operation to extract those tuples where Client.clientNo = Viewing.clientNo. σClient.clientNo=Viewing.clientNo((∏clientNo,fName, lName(Client)) Χ (∏clientNo, propertyNo, comment(Viewing))) Cartesian product and Selection can be reduced to a single operation called a Join.
28. 28. 28 Join Operations Join is a derivative of Cartesian product. Equivalent to performing a Selection, using join predicate as selection formula, over Cartesian product of the two operand relations. One of the most difficult operations to implement efficiently in an RDBMS and one reason why RDBMSs have intrinsic performance problems.
29. 29. 29 Join Operations Various forms of join operation – Theta join – Equijoin (a particular type of Theta join) – Natural join – Outer join – Semijoin
30. 30. 30 Theta join (θ-join) R FS – 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.ai θ S.bi where θ may be one of the comparison operators (<, ≤, >, ≥, =, ≠).
31. 31. 31 Theta join (θ-join) Can rewrite Theta join using basic Selection and Cartesian product operations. R FS = σF(R Χ S) Degree of a Theta join is sum of degrees of the operand relations R and S. If predicate F contains only equality (=), the term Equijoin is used.
32. 32. 32 Example - Equijoin List the names and comments of all clients who have viewed a property for rent. (ΠclientNo,fName, lName(Client)) Client.clientNo=Viewing.clientNo (ΠclientNo, propertyNo, comment(Viewing))
33. 33. 33 Natural join R S – 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.
34. 34. 34 Example - Natural join List the names and comments of all clients who have viewed a property for rent. (ΠclientNo, fName,lName(Client)) (ΠclientNo,propertyNo, comment(Viewing))
35. 35. 35 Outer join To display rows in the result that do not have matching values in the join column, use Outer join. R S – (Left) outer join is join in which tuples from R that do not have matching values in common columns of S are also included in result relation.
36. 36. 36 Example - Left Outer join Produce a status report on property viewings. ΠpropertyNo, street, city(PropertyForRent) Viewing
37. 37. 37 Semijoin R FS – Defines a relation that contains the tuples of R that participate in the join of R with S. – It performs a join on two relations and then project Over the attributes of first operand. Can rewrite Semijoin using Projection and Join: R F S = ΠA(R F S)
38. 38. 38 Example - Semijoin List complete details of all staff who work at the branch in Glasgow. Staff Staff.branchNo=Branch.branchNo(σcity=‘Glasgow’(Branch))
39. 39. 39 Division R ÷ S – Defines a relation over the attributes C that consists of set of tuples from R that match combination of every tuple in S. Expressed using basic operations: T1 ← ΠC(R) T2 ← ΠC((S X T1) – R) T ← T1 – T2
40. 40. 40 Example - Division Identify all clients who have viewed all properties with three rooms. (ΠclientNo,propertyNo(Viewing)) ÷ (ΠpropertyNo(σrooms=3 (PropertyForRent)))
41. 41. Relational DBMS The following tables form part of a database held in a relational DBMS: Hotel (hotelNo, hotelName, city) Room (roomNo, hotelNo, type, price) Booking (hotelNo, guestNo, dateFrom, dateTo, roomNo) Guest (guestNo, guestName, guestAddress) 41
42. 42. Exercise – Determine temporary relations 42
43. 43. Exercise – Gererate Relational algebra List all hotels. List all single rooms with a price below £20 per night. List the names and cities of all guests. List the price and type of all rooms at the Grosvenor Hotel. List all guests currently staying at the Grosvenor Hotel. List the details of all rooms at the Grosvenor Hotel, including the name of the guest staying in the room, if the room is occupied. List the guest details (guestNo, guestName, and guestAddress) of all guests staying at the Grosvenor 43
44. 44. SQL Structured Query Language (SQL) – Standardised by ANSI – Supported by modern RDBMSs Commands fall into three groups – Data Definition Language (DLL) » Create tables, etc – Data Manipulation Language (DML) » Retrieve and modify data – Data Control Language » Control what users can do – grant and revoke privileges 44