Relations. Power set Cartesian product Relation Function

Transcription

1 Relations

2 Relations By the end of this part of the course the student should understand and be able to use the concepts of relations and functions in a Z specification. The concepts introduced are: Power set Cartesian product Relation Function

3 A Registry Specification Declarations registered is a set of STUDENTs options is a set of MODULES is_enrolled_on is a relation between STUDENTs and MODULEs Invariant More later, but essentially this is saying that Only STUDENTs who are registered may be enrolled_on MODULEs that are options.

4 Power Set The power set of a set is the set of all subsets of that set. The power set of a set A is written e.g. STUDENT = {Bill, John, Sue}

5 The power set of STUDENT Bill Sue John STUDENT= {Bill, John,Sue} Three Elements Bill John Sue Sue John Sue Bill John Bill Bill Sue John

6 If A has n elements, how many does its powerset have? % 0% 0% 0% 0%

7 Which of the following are in? % 0% 0% 0% 0%

8 Cartesian Product of two sets The cartesian product of two sets A and B is the set of all ordered pairs (x,y), where x is an element of A and y is an element of B The cartesian product of A and B is written A x B Example:

9 How many elements does A x B have? % 0% 0% 0%

10 Relations A Relation, ρ, from a set A to a set B, is a subset of AXB. It is a set of pairs. is_on = {(john,cs189), (sue,cs189), (sue,cs154)} knows_about={(john,cs189), (john,cs154), (bill,cs189), (bill,cs154), (sue,cs189), (sue,cs154) } likes={ } attends={(john,cs189), (john,cs154), (sue,cs189), (sue,cs154)}

11 Relations in UML diagrams The existence of a relationship between objects of type A and objects of type B is denoted: A B

12 Relations: sets of ordered pairs (mary, CS266) (john, CS189) (mark, CS189) (john,cs153) (william, CS266) (sue, CS153) (mark, CS153) (mary is_enrolled_on CS266) (john is_enrolled_on CS189) (john is_enrolled_on CS153) (mark is_enrolled_on CS189) (william is_enrolled_on CS266) (sue is_enrolled_on CS153) (mark is_enrolled_on CS153)

13 is_enrolled_on as a picture John Mary Sue Bill Mark William CS153 CS189 CS266 CS183 CS288

14 is_enrolled_on as a database table John John Mary Sue Mark Mark William CS153 CS189 CS266 CS153 CS153 CS189 CS266

15 Infix notation If then we can write This is called infix notation e.g. (Mary,CS266) is_enrolled_on written as Mary is_enrolled_on CS266 e.g. 2 < 3

16 Domain The domain of a relation is the set of all elements in A that are related by to something in B This is written Thus if then It is always true that

17 The set of students enrolled on some course dom(is_enrolled_on) Bill John Mary Sue Mark William CS153 CS189 CS266 CS183 CS288

18 What is dom(r)? 0% 0% 0% 0% 1. {a,b,c} 2. {(a,1),(a,2),(b,4)} 3. {a,b} 4. {1,2,4} a b 1 2 c R 4 3

19 Range The range of a relation is the set of all elements in B that are related by to something in A This is written Thus if then It is always true that

20 The set of courses with some students enrolled ran(is_enrolled_on) John Mary Sue Bill Mark William CS153 CS189 CS266 CS183 CS288

21 What is ran(r)? 0% 0% 0% 0% 1. {1,2,3,4} 2. {1,2} 3. {a,b} 4. None of the above a b 1 2 c R 4 3

22 Library example again is_on_loan_to is a relation from BOOKS to PEOPLE Library a set stock a set members A rule is_on_loan_to is_on_loan_to : BOOKS PEOPLE We will use relational notation to specify the library system

23 The Library state in Z We now have an INVARIANT on the Library state: that only elements of stock can be on loan, and only to members.

24 Borrowing Books Borrow(book b, person p) b is not already on loan to anyone Add b is on loan to p `Delta means change of state } Declare input parameters } Preconditions Postcondition

25 Exercise: returning books Return(book b, person p) b is currently on loan to p Remove b is on loan to p

26 Summary Power set Cartesian Product Relations Infix notation 2<3 Domain of a relation/function Range of a relation/function Z schemas Input and output parameters in Z event schemas [Currie chapters 6-7, Haggarty chapters 4-5 ]

27 More relational notation

28 Running example: registered Abigail Barney Charlotte Dora Emily Fran CS189 CS153 CS154 CS173 CS183

29 Domain restriction

30 registered without domain restriction Abigail Barney Charlotte Dora Emily Fran CS189 CS153 CS154 CS173 CS183

31 registered with domain restriction Abigail Barney Charlotte Dora Emily Fran CS189 CS153 CS154 CS173 CS183

32 Domain anti-restriction

33 Example: Registered with domain anti-restriction Abigail Barney Charlotte Dora Emily Fran CS189 CS153 CS154 CS173 CS183

34 Range restriction

35 Example: registered with a range restriction Abigail Barney Charlotte Dora Emily Fran CS189 CS153 CS154 CS173 CS183

36 Range anti-restriction

37 Example: registered with a range anti-restriction Abigail Barney Charlotte Dora Emily Fran CS189 CS153 CS154 CS173 CS183

38 What is denoted by? 0% 1. The course CS154 0% 0% 2. The people registered for CS Something else

39 What is denoted by? 0% 1. Those of ian, james, kate who have registered for some courses 0% 2. The courses ian, james, and kate have registered for 0% 3. The course registrations that ian, james, and kate are involved in

40 What is denoted by? 1. The people not registered for CS The people registered for courses other than CS The registrations for courses other than CS154 0% The people not re... The people regist... The registrations...

41 Relational inverse

42 Example: inverse of registered CS189 CS153 CS154 CS173 CS183 Abigail Barney Charlotte Dora Emily Fran

43 Relational image

44 Example: Registered Abigail Barney CS189 CS153 Charlotte Dora Emily Fran CS154 CS183 CS173

45 Example: extracting information from a relation

46 Exercise: extracting information

47 What denotes the courses that Ian is registered for? % 0% 0% 0%

48 Summary

49 Please ensure you return your handset before leaving. The handset is useless outside of this class and non-returns will decease the likelihood of future voting system use on this course.