Relations

1
Relations

• Chapter 9
• Formal Specification using Z

2
Formal definition of Relation

• The Z examples so far have been limited to one
  basic type. We need of relating one set to
  another.
• A relation is a set which is based in the idea of
  the Cartesian product.

3
Formal definition of Relation

• A Cartesian product is the pairing of values of
  two or more sets. The Cartesian product is
  written
• X x Y x Z
• For example
• red, green xhardtop,softtop
• (red, hardtop), (red, softtop), (green,
  hardtop), (green,softtop)
• This set consists of ordered paired called a
  tuple

4
Formal definition of Relation

• A binary relation R from the set X to the set Y
  is a subset of the Cartisian Product X Y. If
  (x,y) R, we write xRy and say x is related to
  y. In the case XY, we call R a binary relation
  on X.
• The domain of R is the set
• x X (x,y) R from some y Y
• The range of R is the set
• y Y (x,y) R for some x X

5
Relation as a table

• A relation can be thought of as a table that
  lists the relationship of elements to other
  elements
• Student Course
• Bill Computer Science
• Mary Maths
• Bill Art
• Beth History
• Beth Computer Science
• Dave Math
• The table is a set of ordered pairs

6
Relation as a set of ordered Pairs

• If we let
• X Bill, Mary, Beth, Dave
• Y ComputerScience, Math, Art, History
• then the relation from the prevuous table is
• R (Bill,ComputerScience), (Mary, Math),
  (Bill,Art), (Beth,ComputerScience),
  (Beth,History), (Dave, Math)
• We say Mary R Math. The domain is the domain
  (first column) of R is the set X the range
  (second column) of R is the set Y. So far the
  relation has been defined by specifying the
  ordered pairs involved.

7
Relation defined by rule

• It is also possible to define a relation by
  giving a rule for membership of the relation.
• Let X 2,3,4 and Y 3,4,5,6,7
• If we define R from X to Y by
• (x,y) R if x divides y (with zero remainder)
• We get
• R (2,4), (2,6), (3,3), (3,6), (4,4)

8
Relation written as table

• This can be written as a table .
• X Y
• 2 4
• 2 6
• 3 3
• 3 6
• 4 4
• The domain of R is the set 2,3,4 and the range
  is 3,4,5

9
Let R be the relation on X 1,2,3,4 defined
by (x,y) R if x y, x,y X. is shown
The digraph for R
1
2
3
4
R (1,1), (1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3
,3),(3,4),(4,4) The domain and the range are
both equal to x. R is reflexive, transitive and
antisymmetric.
10
Let R be the relation on X 1,2,3,4 defined
by (x,y) R if x lt y, x,y X. is shown
The digraph for R
1
2
3
4
R (1,2),(1,3),(1,4),(2,3),(2,4),(3,4) The
domain and the range are both equal to x. R is
reflexive, transitive and non-symmetric.
11
Types of relation

• A relation R on a set X is reflexive if
  (x,x) R for every x X.
• A relation R on a set X is symmetric if for all
  x,y X, if (x,y) R, then (y,x) R.
• A relation R on a set X is antisymmetric if for
  all x,y X, if (x,y) R, (y,x) and x y
  , then (y,x) R.

12
Types of relation

• A relation R on a set X is transitive if for all
  x,y,z X, if (x,y) R, and (y,z) R
  then (x,z) R.

13
Types of relation

• Let ? be a relation on the set A, then
• ? is reflexive if x? x for all x? A
• ? is symmetric if x? y implies y ? x
• ? is transitive if x? y and y? z ? imply x? z
• ? is antisymmetric if x? y and y? x imply that xy

14
Relations

• A special case of Cartesian product is a pair. A
  binary relation is set of pairs, of related
  values.
• COUNTRY the set of all countries
• LANGUAGE set of all languages

15
A relation on COUNTRY and LANGUAGE called SPEAKS
could have values (France,French),(Canada,French
),(Canada,English),(GB,English),(USA,English)
SPEAKS relation
France Canada GB USA
French English
LANGUAGE
COUNTRY
16
RELATIONS

• As can be seen from the SPEAKS relation, there
  are no restrictions requiring values of the two
  sets to be paired only one-to-one.
• We are free to choose the direction or order of
  the relation
• SPOKEN ? (LANGUAGE ?COUNTRY)

17
Declaring Relations in Z

• R X ? Y ( equivalent to ?(X ?Y) )
• SPEAKS COUNTRY ? LANGUAGE
• ( equivalent to ?(COUNTRY ? LANGUAGE) )
• In English we say
• R relates X to Y.
• SPEAKS relates COUNTRY to LANGUAGE

18
MAPLETS

• A maplet relates x to y and is written
• x ?y (x,y)
• x is related to y or x maps to y
• GB ? English ? SPEAKS
• is equivalent to
• (GB, English) ? SPEAKS

19
Membership

• To discover if a certain pair of values are
  related it is sufficient to see if the pair or
  maplet is an element of the relation
• GB ? English ? SPEAKS
• OR
• (GB, English) ? SPEAKS

20
Infix Notation

• If we declare a relation using low-line
  characters as place holders to show the fact that
  is infix
• _SPEAKS_ COUNTRY ? LANGUAGE
• GB speaks English
• In general
• xRy x ? y ? R(x,y) ? R

21
The DOMAIN and RANGE of the relation SPEAKSdom
SPEAKS is set of countries where at least one
language is spoken.ran SPEAKS is set of
languages which are spoke in at least one
country.
SPEAKS relation
French English
France Canada GB USA
LANGUAGE (TARGET)
COUNTRY (SOURCE)
dom SPEAKS ran
SPEAKS
22
Domain and Range

• A relation relates values of a set called source
  or from-set to a set called target or to-set.
• Below the source is X and the target Y.
• ?R X? Y
• ?______________________
• ? R (x1,y2), (x2,y2), (x4,y3), (x6, y3) ),
  (x4,y2)
• ?________________________

23
Domain

• In most cases only a subset of of the source is
  involved in the relation. This subset is called
  the domain
• dom R x1,x2,x4,x6

24
Range

• In most cases only a subset of of the target is
  involved in the relation. This subset is called
  the range
• ran R y2,y3

25
The DOMAIN and RANGE of the relation R dom R
x1,x2,x4,x6 ran R y2,y3
R relation
y1
x5
x1 x2 x4 x6
Y2 y3
y4
Y (TARGET)
X (SOURCE)
x3
dom R ran R
26
Relational Image

• To discover the set of values from the range of a
  relation related to a set of values from the
  domain of the relation one can use the relational
  image
• R?S?
• pronounced the relational image of S in R. For
  the languages spoken in France and Switzerland
  would be
• SPEAKS ?France, Switzerland?
• French,German, Italian, Romansch

27
Constant value for a Relation

• Some relations have constant values. If the value
  of a relation is known, it can be given by an
  axiomatic definition. For example the infix
  relation greater-than-or-equal
• ?_ ? _ ???
• ?______________
• ? ? i,j? ? i ? j ??n? ? ij n
• ?________________

28
Domain Restriction

• The domain restriction operator restricts a
  relation to that part where the domain is
  contained in a particular set. The relation R
  domain restricted to S, is written as
• S ? R
• The point of the operator can be thought of as
  pointing left to the domain

29
Range Restriction

• The range restriction operator restricts a
  relation to that part where the range is
  contained in a particular set. The relation R
  range restricted to S, is written as
• R ? S
• The point of the operator can be thought of as
  pointing right to the range

30
Domain Subtraction

• The Domain Subtraction operator restricts a
  relation to that part where the domain is not
  contained in a particular set. The relation R
  domain subtracted to S, is written as
• S ? R
• The point of the operator can be thought of as
  pointing left to the domain

31
Range Subtraction

• The Range Subtraction operator restricts a
  relation to that part where the Range is not
  contained in a particular set. The relation R
  range subtracted to S, is written as
• R ? S

32
Example of Domain Restriction

• ?Hols_____
• ?holidays COUNTRY ?DATE
• ?__________
• EU ?COUNTRY
• The relation mapping only EU countries to their
  public holidays is
• EU ? holidays
• The relation mapping only non-EU countries to
  their public holidays is
• EU ? holidays

33
Composition

• Relations can be joined together in an operation
  called composition. Given
• R X ? Y and
• Q Y ? Z
• We can have forward composition or backward
  composition

34
Forward Composition

• Given
• R X ? Y and
• Q Y ? Z
• forward composition is defined as
• RQ X ? Z

35
Forward Composition
RQ
R
Q
X
Z
Y
36
For any pair (x,y) related by R forward composed
with Q x RQ z there is a y where R relates x to
y and Q relates y to z ?yY ? xRy ? yQz
RQ
R
Q
X
Z
Y
37
Backward Composition

• Given
• R X ? Y and
• Q Y ? Z
• backward composition is defined as
• Q?R RQ

38
Repeated Composition

• A homogeneous relation is one which relates
  values from a type to values of the same type.
  Such a relation can be composed with iteslf
• R X ? X
• R R X ? X

39
Composition Example

• Countries are related by the borders relation if
  the share a border
• borders COUNTRY ? COUNTRY
• e.g. France borders Switzerland
• Switzerland borders Austria
• Countries are related by borders composed with
  borders if they each share a border with a third
  country.

40
Composition Example

• The two facts
• 1. France borders Germany
• 2. Germany borders Austria
• can be represented by
• France bordersborders Austria
• We can write
• Spain borders3 Denmark
• for
• Spain borders borders borders Denmark

41
Transitive Closure

• In general the transitive closure R as used in
• x R y
• means that there is a repeated composition of R
  which relates x to y. For example
• France borders India.
• Means France is on the same land-mass as India

42
Identity Relation

• The identity relation
• id X
• is the relation which maps all xs on to
  themselves
• id X xX ? x ? x

43
Reflexive Transitive Closure

• The repeated composition
• R x R ? id X
• includes the identity relation. The Reflexive
  Transitive Closure is similar to the Transitive
  Closure except that it includes the identity
  relation. For example
• x R x is true, even if x R x is false, (id x)
• France borders France is true

44
Relation Example

• Public holidays around the world can be described
  as follows
• COUNTRY the set of all countries
• DATE the dates of a given year
• ?Hols_____
• ?holidays COUNTRY ?DATE
• ?__________

45
Relation Example

• An operation to discover whether a date d? is a
  public holiday in country c? is
• REPLY yes no
• ?Enquire_____
• ??Hols
• ?c? COUNTRY
• ?d? DATE
• ?rep! REPLY
• ?________ ?
• ?(c? ? d? ? holidays ? rep! yes) ?
• ?(c? ? d? ? holidays ? rep! no)
• ?__________

46
Relation Example

• An operation to decree whether a public holiday
  in country c? on date d? is
• ?Decree_____
• ??Hols
• ?c? COUNTRY
• ?d? DATE
• ?________ ?
• ?holidays holidays ? c? ? d?
• ?__________

47
Relation Example

• An operation to abolish a public holiday in
  country c? on date d? is
• ?Abolish_____
• ??Hols
• ?c? COUNTRY
• ?d? DATE
• ?________ ?
• ?holidays holidays \ c? ? d?
• ?__________

48
Relation Example

• An operation to find the dates ds! of all public
  holidays in country c? is
• ?Dates_____
• ??Hols
• ?c? COUNTRY
• ?ds? ?DATE
• ?________ ?
• ?ds! holidays ?c??
• ?__________

49
Inverse of a relation

• The inverse of a relation R from X to Y
• R X ? Y is written
• R
• and it relate Y to X (mirror image). So if
• x R y
• then
• y Rx

50
Family Example

• PERSON the set of all persons
• father,mother PERSON ? PERSON
• x father y meaning x has father y
• v mother w meaning v has mother w

51
Family Example parent relation

• parent PERSON ? PERSON
• can be defined as
• parent father ? mother

52
Family Example sibling

• sibling PERSON ? PERSON
• can be defined as
• sibling (parent parent ) \ id PERSON
• This is defined as the relation parent composed
  with its own inverse, in other words the set of
  persons with the same parent. Note the inverse of
  parent that relates parent to child. A person is
  not usually counted as their own sibling, hence
  the use of the identity relation.

53
sibling PERSON ? PERSON can be defined
as sibling (parent parent ) \ id
PERSON children parents ?PERSON children
Pat, Bernard,Marion parents Liam, Mary
parentparent
R
Q
Pat Bernard Marion
Liam Mary
Pat Bernard Marion
children
Z
Y
54
(parent parent ) (Pat, Pat),
(Pat,Bernard), (Pat,Marion), (Bernard, Bernard),
(Bernard,Pat),
(Bernard,Marion), (Marion,Marion),
(Marion,Pat),
(Marion,Bernard) (parent parent ) \ id
PERSON (Pat,Bernard), (Pat,Marion),(Bernard,P
at),
(Bernard,Marion), (Marion,Pat),

(Marion,Bernard)
parentparent
parent
parent
Pat Bernard Marion
Liam Mary
Pat Bernard Marion
children
Z
Y
55
Family Example ANCESTOR

• ancestor PERSON ? PERSON
• can be defined as
• ancestor parent
• The relation ancestor can be defined as the
  repeated composition of parent.

56
Chapter 9 Question 1

• Express the fact that the language Latin is not
  the official language of any country

57
Chapter 9 Question 1

• Express the fact that the language Latin is not
  the official language of any country
• LatinLANGUAGE
• Latin ? ran speaks

58
Chapter 9 Question 2

• Express the fact that Switzerland has four
  languages

59
Chapter 9 Question 2

• Express the fact that Switzerland has four
  languages
• speaks ? Switzerland ? 4

60
Chapter 9 Question 3

• Give a value to the relation speaksInEU which
  relates countries which are in the set EU to
  their languages.

61
Chapter 9 Question 3

• Give a value to the relation speaksInEU which
  relates countries which are in the set EU to
  their languages.
• EU ? COUNTRY
• speaksInEU COUNTRY ? LANGUAGE
• speaksInEU EU ? speaks

62
Chapter 9 Question 4

• Give a value to the relation grandParent

63
Chapter 9 Question 4

• Give a value to the relation grandParent
• grandParent PERSON ? PERSON
• grandParent parent parent

64
Chapter 9 Question 5

• A persons first cousin is defined as a child of
  the persons aunt or uncle. Give a value for the
  relation firstCousin.

65
Chapter 9 Question 5

• A persons first cousin is defined as a child of
  the persons aunt or uncle. Give a value for the
  relation firstCousin.
• firstCousin PERSON ? PERSON
• firstCousin (grandParent grandParent ) \
  sibling

66
Chapter 9 Question 6

• Given
• PERSON set of uniquely identifiable persons
• MODULE set module numbers at a university
• students,teachers,EU,inter ? PERSON
• offered ? MODULE
• studies PERSON ? MODULE
• teaches PERSON ? MODULE

67
Chapter 9 Question 6

• PERSON set of uniquely identifiable persons
• MODULE set module numbers at a university
• students,teachers,EU,inter ? PERSON
• offered ? MODULE
• studies PERSON ? MODULE
• teaches PERSON ? MODULE
• Explain the predicates
• EU ? inter ?
• EU ? inter students
• dom studies ? students
• dom teaches ? teachers
• ran studies ? offered
• ran teaches ran studies

68
Chapter 9 Question 6

• Explain the effect of each predicates
• EU ? inter ?
• EU ? inter students
• dom studies ? students
• dom teaches ? teachers
• ran studies ? offered
• ran teaches ran studies
• Students are either from the EU or overseas, but
  not from both.
• Students study and teachers teach.
• Only offered modules can be studied.
• Those modules that are taught are studied.

69
Chapter 9 Question 7

• Give an expression for the set of modules studied
  by a person p.

70
Chapter 9 Question 7

• Give an expression for the set of modules studied
  by a person p.
• studies ? p ?

71
Chapter 9 Question 8

• Give an expression for the number of modules
  taught by a person q.

72
Chapter 9 Question 8

• Give an expression for the number of modules
  taught by a person q.
• (teaches ? q ? )

73
Chapter 9 Question 9

• Explain what is meant by the inverse of the
  relation studies
• studies

74
Chapter 9 Question 9

• Explain what is meant by the inverse of the
  relation studies
• studies
• The inverse of the relation studies relates
  modules to persons studying them.

75
Chapter 9 Question 10

• Explain what is meant by the composition of the
  relations
• studies teaches

76
Chapter 9 Question 10

• Explain what is meant by the composition of the
  relations
• studies teaches
• The composition of relates students to the
  teachers who teach modules the students study.

77
Chapter 9 Question 11

• Give an expression for the set of persons who
  teach person p.

78
Chapter 9 Question 11

• Give an expression for the set of persons who
  teach person p.
• (studies teaches)? p ?

79
Chapter 9 Question 12

• Give an expression for the number of persons who
  teach both person p and person q.

80
Chapter 9 Question 12

• Give an expression for the number of persons who
  teach both person p and person q.
• ((studies teaches)? p ? ?(studies
  teaches)? q ?)

81
Chapter 9 Question 13

• Give an expression for that part of the relation
  studies that pertains to international
  students(inter)

82
Chapter 9 Question 13

• Give an expression for that part of the relation
  studies that pertains to international
  students(inter)
• inter ? studies

83
Chapter 9 Question 14

• Give an expression for that states that p and q
  teach some of the same international students.

84
Chapter 9 Question 14

• Give an expression for that states that p and q
  teach some of the same international students.
• ((teaches studies )? p ? ? (teaches studies
  ) ? q ?) ? inter ? ?

85
Chapter 9 Question 15

• The following declarations are part of the
  description of an international conference.
• PERSON set of uniquely identifiable persons
• LANGUAGE the set of world languages
• official ? LANGUAGE

86
Chapter 9 Question 15

• The following declarations are part of the
  description of an international conference.
• PERSON set of uniquely identifiable persons
• LANGUAGE the set of world languages
• official ? LANGUAGE

87
Chapter 9 Question 15

• PERSON set of uniquely identifiable persons
• LANGUAGE the set of world languages
• official ? LANGUAGE
• ?CONFERENCE_____
• ? ? PERSON
• ? ? LANGUAGE
• ? _speaks_ PERSON ? LANGUAGE
• ?__________

88
Chapter 9 Question 15A

• Write expression that states every delegate
  speaks at least one language
• delagates ? dom speaks

89
Chapter 9 Question 15B

• Write expression that states every delegate
  speaks at least one official language of the
  conference
• delagates ? ran speaks ? ?

90
Chapter 9 Question 15C

• Write expression that states every delegate
  speaks at least one official language of the
  conference
• ?langLANGUAGE ?
• (?delPERSON del? delagates ? del speaks lang)

91
Chapter 9 Question 15E

• Write an operation schema to register a new
  delegate and the set of languages he or she
  speaks.
• ?Register_____________
• ? del?PERSON
• ? lang? ?LANGUAGE
• ?? CONFERENCE
• ?________________
• ?del? ? delegates
• ?delegates delegates ? del?
• ?speaks
• ?speaks ? lanLANGUAGE lan? lang? ? del? ?lan
• ?official official
• ?_________________

92
Labs

• Using pen and paper do Chapter 2 Exercises 5.
• Using pen and paper do exam example Edward,
  Fleur, and Gareth.
• Type both of the above into Word using symbols or
  Z fonts.

93
Labs

• Calculate the following unions.
• S1 1,2, S2 2, 3, 4, 5, S3 1, 2, 2, 3,
  4, 5, union(S1,S2,W).
• S1 1,2, S2 2, 3, 4, 5, S3 1, 2, 2, 3,
  4, 5, union(S1,S3,W).
• S1 1,2, S2 2, 3, 4, 5, S3 1, 2, 2, 3,
  4, 5, union(S3,S2,W).

94
Labs

• Start Prolog
• Copy sets.pro from X\distrib\wmt2\maths\prolog
• Consult sets.pro
• Use the sets programs to solve the previous union
  of S1,S2 and S3.

95
Labs

• Using prolog do exam example Edward, Fleur, and
  Gareth.

96
Labs

• 1)Using pen and paper do Chapter 2 Exercises 5.
• 2)Using pen and paper do exam example Edward,
  Fleur, and Gareth.
• 3)Calculate the following unions.
• S1 1,2, S2 2, 3, 4, 5, S3 1, 2, 2, 3,
  4, 5, union(S1,S2,W).
• S1 1,2, S2 2, 3, 4, 5, S3 1, 2, 2, 3,
  4, 5, union(S1,S3,W).
• S1 1,2, S2 2, 3, 4, 5, S3 1, 2, 2, 3,
  4, 5, union(S3,S2,W).
• 4)Type both 1 and 2 above into Word using symbols
  or Z fonts.
• 5)Start Prolog
• 6)Copy sets.pro from X\distrib\wmt2\maths\prolog
  
• 7)Consult sets.pro
• 8)Use the sets programs to solve the previous
  union of S1,S2,S3
• 9)Using prolog do exam example Edward, Fleur, and
  Gareth.