Relations Relations By the end of this part of the course the - - PDF document

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Relations Relations By the end of this part of the course the - - PDF document

Nov 27, 2022 277 likes •774 views

Relations 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

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Relations

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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

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A “Registry” Specification

Declarations registered is a set of STUDENTs

  • ptions 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.

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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}

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Three Elements

The power set of STUDENT

John Bill Sue John Bill Sue John Bill Sue John Bill Bill Sue Sue John

STUDENT= {Bill, John,Sue}

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

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1. 2. 3. 4. 5.

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Which of the following are in ?

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1. 2. 3. 4. 5.

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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:

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How many elements does A x B have?

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1. 2. 3. 4.

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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)}

  • Relations
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Relations in UML diagrams

The existence of a relationship between

  • bjects of type A and objects of type B

is denoted:

A B

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(mark is_enrolled_on CS153) (mary is_enrolled_on CS266) (sue is_enrolled_on CS153) (william is_enrolled_on CS266) (mark is_enrolled_on CS189) (john is_enrolled_on CS153) (john is_enrolled_on CS189) (mark, CS153) (mary, CS266) (sue, CS153) (william, CS266) (mark, CS189) (john,CS153) (john, CS189)

Relations: sets of ordered pairs

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is_enrolled_on as a picture

John Mary Sue Mark William Bill CS189 CS153 CS266 CS183 CS288

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is_enrolled_on as a database table

John CS153 John CS189 Mary CS266 Sue CS153 Mark CS153 Mark CS189 William CS266

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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

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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

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dom(is_enrolled_on)

John Mary Sue Mark William Bill CS189 CS153 CS266 CS183 CS288 The set of students enrolled on some course

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What is dom(R)?

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a b c 1 2 3 4

1.

{ a,b,c}

2.

{ (a,1),(a,2),(b,4)}

3.

{ a,b}

4.

{ 1,2,4}

R

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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

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ran(is_enrolled_on)

John Mary Sue Mark William Bill CS189 CS153 CS266 CS183 CS288 The set of courses with some students enrolled

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What is ran(R)?

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a b c 1 2 3 4 R

1.

{ 1,2,3,4}

2.

{ 1,2}

3.

{ a,b}

4.

None of the above

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Library example again

is_on_loan_to is a

relation from BOOKS to PEOPLE

is_on_loan_to : BOOKS PEOPLE We will use relational notation to specify the

library system

Library a set stock a set members A rule is_on_loan_to

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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.

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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

}

}

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Exercise: returning books

Return(book b, person p) b is currently on loan to p Remove b is on loan to p

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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 ]

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More relational notation

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Running example: registered

Abigail Barney Charlotte Dora Emily CS189 CS153 CS154 CS173 CS183 Fran

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Domain restriction

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registered without domain restriction

Abigail Barney Charlotte Dora Emily CS189 CS153 CS154 CS173 CS183 Fran

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registered with domain restriction

Abigail Barney Charlotte Dora Emily CS189 CS153 CS154 CS173 CS183 Fran

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Domain anti-restriction

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Abigail Barney Charlotte Dora Emily CS189 CS153 CS154 CS173 CS183 Fran

Example: Registered with domain anti-restriction

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Range restriction

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Example: registered with a range restriction

Abigail Barney Charlotte Dora Emily CS189 CS153 CS154 CS173 CS183 Fran

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Range anti-restriction

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Example: registered with a range anti-restriction

Abigail Barney Charlotte Dora Emily CS189 CS153 CS154 CS173 CS183 Fran

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What is denoted by ?

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1.

The course CS154

2.

The people registered for CS154

3.

Something else

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What is denoted by ?

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1.

Those of ian, james, kate who have registered for some courses

2.

The courses ian, james, and kate have registered for

3.

The course registrations that ian, james, and kate are involved in

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What is denoted by ?

0% 0% 0%

The people not re... The people regist... The registrations...

1.

The people not registered for CS154

2.

The people registered for courses other than CS154

3.

The registrations for courses other than CS154

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Relational inverse

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Example: inverse of registered

Abigail Barney Charlotte Dora Emily CS189 CS153 CS154 CS173 CS183 Fran

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Relational image

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Example: Registered

Abigail Barney Charlotte Dora Emily CS189 CS153 CS154 CS173 CS183 Fran

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Example: extracting information from a relation

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Exercise: extracting information

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What denotes the courses that Ian is registered for?

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1. 2. 3. 4.

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Summary

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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.