Set Theory Introduction We introduce the use of Set Theory in - PDF document

Set Theory

Introduction � We introduce the use of Set Theory in specifying computer systems � We introduce: Z schemas for defining states and � events/relationships between states. � Some basic set notation. “is (not) an element of”, � “Set Union”, � “Set Difference” �

System Lifecycle Requirements analysis � System Specification � Design � Implementation � Testing and Integration � Support and Enhancement �

Abstraction?! Abstract Representation 1933 Geographical Representation 1932 Maps from London Underground , Idea from Woodcock and Davies

Scenario � Your software company has been approached to design and build a software system to manage the book loans in a library.

Exercise: Library system � We want to describe a system to track the borrowing of books from a library. � Identify three elements of a real library system that you will want to retain in your description. � Identify six elements of a real library system that you will want to abstract away. � What kind of things should your specification describe?

Informal Library System STATE Acquire( book b) Library b is not already in the set stock a set stock Add b to the set stock a set members A rule is_on_loan_to Dispose( book b) EVENTS Join( person p) p is not already in the set members Leave( person p) Add p to the set members

Borrowing and returning books Library a set stock a set members A rule is_on_loan_to Borrow( book b, person p) b is not already on loan to anyone Return( book b , person p) Add b is on loan to p

Towards a Formal Description: Library a set stock a set Members A link is_on_loan_to Acquire( book b) b ∉ stock b is not in the set stock Add b to the set stock Dispose( book b) b is in the set stock b ∈ stock Remove b from the set stock

Changing State 1: Acquire Library a set stock a set Members A link is_on_loan_to Add b to the set stock After the event stock is what stock was before the event Acquire( book b) with b added b ∉ stock { } ′ = = ∪ \ Book s Books b ' { } stock stock b

Changing State 2: Dispose Library a set stock a set members A link is_on_loan_to Remove b from the set stock After the event stock is what stock was before the event Dispose( book b) with b removed b ∈ stock { } ' = \ stock stock b

Exercise: Complete Join and Leave . Library a set stock a set members A link is_on_loan_to Join( person p) p is not already in the set Members Add p to the set Members Leave( person p) p is in the set Members Remove p from the set Members

Toward a Formal Description: Borrow ing and return ing Borrow( book b, person p) b is not already on loan to anyone Library Add b is on loan to p a set stock a set members A link is_on_loan_to Return( book b , person p) Question: What is “is_on_loan_to”?

Summary � Software life cycle � Specification � Sets a ∈ A � a is an element of A : a ∉ A � a is not an element of A : A ∪ B � The Union of two sets A and B : � The set difference of two sets A and B: A \ B

Sets: Basics

Sets: Basics In this lecture we introduce: The basic concept of a set � Notations for denoting sets � Set enumeration � Set comprehension � Basic set operations � Union, intersection, set difference � Use of Venn Diagrams � [ Haggarty chapter 3] and [Currie chapter 3]

Sets: Basic Concept, Notation and Constructions b g r u o u n p c h � What is a set? set g n i r e h t a g d r class e h f a m i l y collection ensemble We could have sets of……………? Some examples of sets of “numbers”… [c.f. Haggerty p36] N = The set of Natural Numbers { 1,2,3,…} Z = The set of Integers { …-3,-2,-1,0,1,2,3,…} Q = The set of Rational Numbers { p/q : p,q integers, not(q= 0) } R = The set of Real Numbers all decimals?

Notation � “A set is a collection of objects called elements” [Haggarty p34] � “A set is a collection of distinct objects called elements or members” [Currie p20] {1,2,3,4} W: set of wives of Henry A(x): x is called Anne The set of wives of {Anne of Cleves, {x ∈ W | A(x)} Henry VIII called Anne Anne Boleyn} The set of all natural numbers {x ∈ {2,4,6,8,…} N |" x is even" } that are even {2,4,6,8,…,2n,…} The set of all natural numbers {x ∈ N |" x is prime" } {2,3,5,7,11,…} that are prime

{x ∈ Exercise: A | B(x)} � Enumerate the following sets using set brackets as on the previous slide. You may need to use dots ∈ < = 2 � E.g. {x Z | x 10} {-3,-2,-1, 0,1,2,3} 2 = ∈ � {x Z | x 1} ∈ < < � {x Z | 0 x and x 5} � {x ∈ months of the year | x does not contain th e letter " r" } {x ∈ � Z | x is double another integer}

Venn Diagrams � A convenient way of visualising/illustrating sets is the use of Venn Diagrams using circles to represent sets and shading to illustrate/distinguish areas of interest. A Venn Diagram A Venn Diagram illustrating … illustrating a set A with… A ∪ The Union of two sets B ∈ ∉ and x A y A y A A ∪ B x A B

Basic Operations on Sets A A B A ∪ A ∩ B B ∈ ∪ x A B ∈ ∩ x A B if and only if if and only if ∈ ∈ x A or x B ∈ ∈ nd x A a x B A B A B A \ B \ B A ∈ \ x A B if and only if A B A B ∈ ∉ nd x A a x B

Subset Set A is a subset of set B if every element of A is an element of B ⊆ " is a subset of " is denoted A B A B A ⊆ B Examples { } { } ⊆ 2 , 3 , 4 1 , 2 , 3 , 4 , 5 , 6 , 7 { } ⊆ B 1 , 2 , 3 , 4 , 5 , 6 , 7 N A { } ⊆ May, June, July Months ⊆ ∪ A A B ∩ ⊆ A B A

A ∩ What symbol is being used in ? B Union 1. Intersection 2. Subset 3. Member of 4. Set difference 5. 0% 0% 0% 0% 0% n f e n t o e o c o s n i r i n t b e c e b U u e r m e S s f r f e e i d M t n t I e S

A ⊆ What symbol is being used in ? B Union 1. Intersection 2. Subset 3. Member of 4. Set difference 5. 0% 0% 0% 0% 0% n f e n t o e o c o s n i r i n t b e c e b U u e r m e S s f r f e e i d M t n t I e S

a ∈ What symbol is being used in ? B Union 1. Intersection 2. Subset 3. Member of 4. Set difference 5. 0% 0% 0% 0% 0% n f e n t o e o c o s n i r i n t b e c e b U u e r m e S s f r f e e i d M t n t I e S

Use of a Sequence of Venn diagrams ∪ = ∩ To show \ ( ) ( \ ) ( \ ) A B C A B A C A \ B A B A \ C B ∪ C C ( ) ( ) ∩ ( ) \ \ A B A C ∪ \ A B C ∪ ∩ \ ( ) ( \ ) ( \ ) The shaded areas for and A B C A B A C ∪ = ∩ are the same. Thus \ ( ) ( \ ) ( \ ) A B C A B A C

Venn Diagram Exercise ∩ = ∪ Is it true that \ ( ) ( \ ) ( \ ) ? A B C A B A C ∪ ∩ = ∪ ∩ Is it true that ( ) ( ) ? A B C A B C ∩ ∪ ⊆ ∩ ∪ Is it true that ( ) ( ) ? A B C A B C

∩ = ∪ Is it true that \ ( ) ( \ ) ( \ ) ? A B C A B A C Yes 1. No 2. Don’t know 3. 0% 0% 0% s o w e N o Y n k t ’ n o D

∪ ∩ = ∪ ∩ Is it true that ( ) ( ) ? A B C A B C Yes 1. No 2. Don’t know 3. 0% 0% 0% s o w e N o Y n k t ’ n o D

∩ ∪ ⊆ ∩ ∪ Is it true that ( ) ( ) ? A B C A B C Yes 1. No 2. Don’t know 3. 0% 0% 0% s o w e N o Y n k t ’ n o D

Size of set � Notation: # A denotes the number of elements in the set A. � E.g. � � �

0% e v o b a e h t f 0% o e n 3 o N 0% 2 0% 1 None of the above B 1. 1 3. 3 2. 2 A 4.

Membership tables 1 ( ) ( ) ( ) ∩ ∪ = ∩ ∪ ∩ Example : To show we must show that A B C A B A C ( ) ∩ ∪ whenever is an element of x A B C ( ) ( ) ∩ ∪ ∩ then it is also a member of A B A C and vice - versa. We can do this by considerin g the various possibilit ies for ' s member ship of and . x A, B C ∉ ∈ ∉ For example if , and then we may deduce x A x B x C ( ) ( ) ( ) ( ) ∉ ∩ ∉ ∩ ∉ ∩ ∪ ∩ , and x A B x A C x A B A C ( ) ( ) ∈ ∪ ∉ ∩ ∪ and . x B C x A B C We need to check this for all eight cases. We can do this in tabular form...

Membership tables 2 ( ) ( ) ( ) ∩ ∪ ∩ ∪ ∩ B ∪ A ∩ A ∩ A B C A B A C C C C A B B 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 ∈ ∉ In the table we place 1 under if and 0 if . SET x SET x SET