Computing Fundamentals 1 Lecture 7 Relations

1
Computing Fundamentals 1 Lecture 7Relations

• Lecturer Patrick Browne
• http//www.comp.dit.ie/pbrowne/
• Room K308
• Based on Chapter 14.
• A Logical approach to Discrete Math
• By David Gries and Fred B. Schneider

2
Relations and Functions

• Tuples are record like structure e.g. ltname,agegt
• Cross products represent the types (or sorts in
  CafeOBJ) in a tuple e.g. String ? Nat
• Relations are relationships between objects.
• Functions are relations with some special
  properties.

3
Tuples1

• Given expressions b and c then ?b,c? is called an
  ordered pair. Well known ordered pairs are ?x,y?
  coordinates in the plane, ?name,address?.
• Axiom, Pair equality
• ?b,c? ?b,c? ? b b ? c c
• A 2-tuple is called an ordered pair.

4
Cross Products

• The Cross product (or Cartesian product) S?T of
  two sets S and T is the set of all ordered pairs
  ?b,c? such that b is in S and c is in T.
• Axiom Cross product (used in later proof)
• S?T b,c b?S ? c?T ?b,c?
• Cross products can be extended to n sets
  producing n-tuples.

5
Cross Products

• A 2-tuple
• S 2,5
• T 1,2,3
• S?T ?2,1?,?2,2?,?2,3?,?5,1?,?5,2?,?5,3?

6
Cross Products()

• Example of a 2-tuple
• S Mary,John
• T John,Bill,Brent
• S?T ?Mary, John?,? Mary,Bill ?,
• ?Mary,Brent ?,? John, John ?,
• ?John,Bill ?,? John,Brent?

7
Cross Products Example()

• SxT / TXS
• S Mary,John
• T John,Bill,Brent
• T?S ltJohn,Marygt,ltJohn,Johngt,ltBill,Marygt,ltBill,J
  ohngt, ltBrent,Marygt,ltBrent,Johngt

8
Some Theorems for Cross products

• Membership
• ?x,y? ? S?T ? x?S ? y?T
• Distributivity of ? over ?
• S ? (T?U) ? (S ? T) ? (S ? U)
• (S ? T) ? U ? (S ? U) ? (T ? U)
• Monotonicity
• T ? U ? (S ? T) ? (S ? U)

9
Relations

• A relation is a subset of a cross product B1? B2
  ? ? Bn
• A binary relation over B?C is a subset of B?C.
• It is possible to have a relation on B?B.
• Well know relation lessThan(i,j).
• Difficult to enumerate (or list), so use
• i,j ? j-i is positive ?i,j?

10
Relations

• A relation can be represented by
• A Set of ordered pairs
• A table e.g. a tuple in a database
• A directed graph
• A set comprehension a rule based description
  similar to the quantifications of Chapter 8
• Two notations for membership of relation ?
• ?b,c? ? ? (set of pairs)
• b ? c (infix, conjunctional)

11
Relation as Directed Graph
One vertex for each element of the set. There is
a directed edge from each vertex b to vertex c
iff ltb,cgt is in the binary relation.
2
1
3
6
4
5

• Set 1,2,3,4,5,6
• Relation lt1,1gt,lt1,3gt,lt2,5gt,lt2,1gt, lt5,3gt

12
Diagraph of Relations

• Given the relation
• R lt1,2gt,lt2,1gt,lt3,3gt,lt1,1gt,lt2,2gt
• On the set
• 1,2,3
• Below is the digraph of R.

1
2
3
13
Digraph

• Here are digraphs of some relations
• P lt1, 4gt ,lt2,1gt, lt3, 2gt
• Q lt1, 4gt ,lt2, 4gt

2
2
1
1
3
4
4
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Digraphs

• Here are digraphs of some relations
• (c) R lt1, 4gt, lt2, 3gt, lt3, 3gt,lt2, 1gt
• (d) S lt3, 1gt ,lt1, 2gt ,lt2, 3gt

2
2
1
1
3
3
4
15
Relations on Sets

• The relation R on 1,2,3,4 is defined by ltx,ygt ?
  R if x2?y .
• The relation R can be written as a set of ordered
  pairs
• R lt1,1gt,lt2,1gt,lt3,1gt,lt4,1gt,lt2,2gt,lt3,2gt,lt4,2gt,lt2
  ,3gt,lt3,3gt,lt4,3gt,lt2,4gt,lt3,4gt,lt4,4gt

16
Some Relations

• The empty relation on S?T is the empty set ?.
• The identity relation iB on B is
• x x ? B ?x,x?
• The relation parent, b is a parent of c.
• The relation predecessor (pred(x)).
• The relation square root (see book1).
• The algorithm relation between input and output
  state.

17
Relations

• Lower case Greek letters represent relations.
• The domain, Dom.?, and range Ran.? of a relation
  ? on B?C are defined as
• Dom.? bB (?c b ? c)
• Ran.? cC (?b b ? c)

18
Relations
?
Dom
Ran
B
C
Dom.? bB (?c b ? c) Ran.? cC
(?b b ? c)
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Operations on Relations

• Let ? be a relation on B?C and ? be a relation
  C?D. The product (or composition) of ? and ?
  denoted ? ? ? is defined by
• ?b,d? ? ??? ? (?c c?C ?b,c??? ? ?c,d???)

???
d
b
c
?
?
D
B
C
Alternative notation b(???)d holds iff b?c?d
holds for some c.
20
Composing Relations
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Powers of a relation

• For example parent?parent can be written parent2.
  
• ?0 iB (the identity relation on B)
• ?n1 ?n ? ? (for n?0)
• ?nm ?n ? ?m
• ?n?m (?n )m

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Classes of relations

• Name Property
• Reflexive (?b b ? b)
• Irreflexive (?b ?(b ? b))
• Symmetric (?b,c (b ? c) ? (c ? b))

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Classes of relations

• Antisymmetric
• (?b,c (b ? c) ? (c ? b) ? bc)
• Asymmetric (non-symmetric, see notes section)
• (?b,c (b ? c) ? ?(c ? b))
• Transitive
• (?b,c,d (b ? c) ? (c ? d) ? b ? d)

24
Relations Discussion

• ? is a relation is on a set, A 1,2,3
• For ? to be either reflexive and irreflexive it
  must hold for all of A.
• For ? to be reflexive then ? must contain the
  following ordered pairs
• lt1,1gt,lt2,2gt,lt3,3gt
• The relation lt (less than) is irrflexive.

25
Relations Discussion

• Consider the relation square
• b square c iff b c c
• square is not reflexive because it does not
  contain lt2,2gt
• square is not irreflexive because it does
  contains pair lt1,1gt
• Thus square neither reflexive or irrflexive. A
  relation that is not reflexive need not be
  irreflexive.

26
Relations Discussion

• For symmetric relations ? does not have to hold
  for all of A, (b ? c) need only hold where (c ?
  b) holds and visa versa.
• The relation ? on 1,2,3
• ? lt1,2gt,lt2,1gt,lt1,3gt
• is neither symmetric nor antisymmetric.
• It is not symmetric because there is no lt3,1gt to
  match the (1,3).
• It is not antisymmetric because it has both lt2,1gt
  and lt1,2gt, but 1?2.

27
Relations Discussion

• The subset (? or ?) relation is antisymmetric.
  If A and B are two sets, and if A is a subset of
  B and B is a subset of A then AB.
• (A ? B) ? (B ? C) ? A B

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Relation Properties()

• S 1 , 2 , 3 .
• R2 lt1,1gt , lt2 ,2 gt , lt3, 3gt , lt3,2gt .
• reflexive, yes lta,agt
• symmetric, no, has lt3,2gt and not lt2,3gt
• Antisymmetric, yes, because it is not symmetric
  and has symmetry on lta,agt.
• Asymmetric (which precludes lta,agt), no, because
  it is antisymmetric (which allows lta,agt).

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Relation Properties()

• S 1 , 2 , 3 .
• R1 lt2,1gt , lt1,2gt , lt2,3gt , lt3,2gt .
• reflexive, no lta,agt
• symmetric, yes for all lta,bgt then ltb,agt
• asymmetric, no, because it is symmetric.
• antisymmetric, no because no lta,bgt,ltb,agt with ab.

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Relation closure Example

• Let R lta,bgt,ltc,agt,ltc,cgt be a relation on the
  set Aa,b,c.
• The reflexive closure is
• r(R) lta,bgt,ltc,agt,lta,agt,ltb,bgt,ltc,cgt
• The symmetric closure is
• s(R) lta,bgt,ltc,agt,ltc,cgt,ltb,agt,lta,cgt

31
Relation Properties

• Let X be the set of all four bit strings. Define
  a relation R on X as lts1,s2gt?R if a substring of
  s1 of length 2 is equal to a substring of s2
  length 2.
• Examples
• lt0111,1010gt ? R
• lt1110,0001gt ? R
• Is this relation reflexive, symmetric,
  antisymmetric, and/or a partial order?

32
Relation Properties

• Is lts1,s2gt?R reflexive?
• Yes. Example lt0111,0111gt ? R
• Is lts1,s2gt?R symmetric?
• Yes. Example
• lt0100,0111gt ? R ? lt0111,0100gt ? R
• Is lts1,s2gt?R antisymmetric?
• No, because it is symmetric.

33
Relation Properties

• Is lts1,s2gt?R transitive? No, here is a counter
  example
• 1000 R 1011 R 0011
• Is lts1,s2gt?R partial order?
• No, because it is not reflexive, not
  antisymmetric, and not transitive. Partial order
  is a property of a relation which is a partial
  order, we will look at later in the lecture.

34
Reflexive or Symmetric?

• On the set 1, 2, 3, 4 we have relation
  (1,1),(1,3),(2,2),(2,4),(3,1),(3,3),(4,2),(4,4)
• We can use a table to check whether the relation
  is reflexive and symmetric.

35
Closure

• The closure of a relation ? with respect to some
  property (e.g. reflexivity) is the smallest
  relation that both has the property and contains
  ?. To construct a closure, add pairs to ?, but
  not too many, until it has the property.
• The less-than relation is not reflexive, but the
  less-than-or-equal-to relation is.
• For example, the reflexive closure of lt over
  integers is the relation constructed by adding to
  the relation all pairs (b,b) for b an integer.
  Therefore ? is the reflexive closure of lt.

36
Sensible Closures

• Reflexive Closure r(?)
• (? ? ?0)
• Symmetric s(?)
• (? ? ?-1)
• Transitive closure ?
• (?1 ? ?2 ? ?3 ? ?4 .. ?n )
• Reflexive Transitive closure ?
• (?0 ? ?1 ? ?2 ? ?3 ? ?4 .. ?n )

37
Sensible Closures

• Let ? be a relation on a set. The reflexive
  (symmetric, transitive) closure of ? is the
  relation ? that satisfies
• ? is reflexive (symmetric, transitive)
• ? ? ?
• If ? is reflexive (symmetric, transitive) and
  ? ? ? then ? ? ? . (e.g. smallest subset)

38
Sensible Closures

• The less-than relation is not reflexive, but
  the less-than-or-equal-to relation is. Yet we
  can form the reflexive closure of less-than
• The reflexive closure of lt over integers is the
  relation constructed by adding to the relation
  all pairs (b,b) for b an integer. Therefore ? is
  the reflexive closure of lt.

39
Closures

• Reflexive closure r(lt) of lt on integers is ?.
• Symmetric closure s(parent) of parent is (parent
  ? child), since if ltb,cgt is in the symmetric
  closure so is ltc,bgt.
• Transitive closure parent of parent is ancestor,
  since if ltb,cgt and ltc,dgt is in the transitive
  closure so is ltb,dgt.
• The reflexive transitive closure parent of
  parent-or-self.

40
Closures

• Let R lt1,2gt,lt3,1gt,lt3,3gt be a relation on the
  set A1,2,3.
• Calculate the following closures
• the reflexive closure,
• the symmetric closure,
• the reflexive transitive closure.

41
Closures

• Reflexive Closure
• r(R) lt1,2gt,lt3,1gt,lt1,1gt,lt2,2gt,lt3,3gt
• Symmetric Closure
• s(R) lt1,2gt,lt3,1gt,lt3,3gt,lt2,1gt,lt1,3gt
• Irreflexive Transitive Closure
• R lt1,2gt,lt3,1gt,lt3,3gt,lt3,2gt
• Reflexive Transitive Closure
• R lt1,2gt,lt3,1gt,lt3,3gt,lt2,2gt,lt1,1gt,lt3,2gt

42
Equivalence Relations

• A relations is an equivalence relation iff it is
  reflexive, symmetric and transitive (e.g. ).
• An equivalence relation ? on a set B partitions
  the set into non-empty disjoint subsets. Elements
  that are equivalent under ? are placed in the
  same partition. Elements that are not equivalent
  under ? are placed in different partitions. For
  example
• ?b,c? ? sameEye ? b and c have same eye colour

43
Equivalence Relations

• Let ? be an equivalence relation on B. Then b?,
  the equivalence class of b, is the subset of
  elements of B that are equivalent (under ?) to b.
• x ? b? ? x ? b
• The sets b? form an equivalence ? relation on B
  partition of B. Thus an equivalence relation on B
  induces a partition of B, where each partition
  element consists of equivalent elements.

44
Equivalence Relations
Objects b and c are in the same partition and
bc
bgreen
.
b
.
cgreen
.
.
c
.
.
.
.
.
d
dblue
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Equivalence Relations
Objects b and c are in the same partition and
bgreencgreen
bgreen
cgreen
dblue
46
Equivalence Classes1
Theorem The equivalence classes of an
equivalence relation R partition the set A into
disjoint nonempty subsets whose union is the
entire set. This partition is denoted A/R and
called, the quotient set, or the partition of A
induced by R, or A modulo R. Definition Let S1,
S2, . . ., Sn be a collection of subsets of A.
Then the collection forms a partition of A if the
subsets are nonempty, disjoint and exhaust A
Si ? ? Si ?Sj ? if i ? j ? Si A
47
Equivalence Relations

• The relation b 4 c on integers 0..9(must be )
• b 4 c ? (b c) is a multiple of 4
• Their difference is a multiple of 4
• We have 4 partitions
• 0 4 8 0,4,8
• 1 5 9 1,5,9
• 2 6 2,6
• 3 7 3,7

48
Equivalence Relations

• Consider the relation ? defined on the set of
  people by
• (b ? c) iff b and c are female and either b and
  c are the same person or b is cs sister.
• Relation ? is reflexive, symmetric, and
  transitive, so it is an equivalence relation. For
  female b, b consists of b and bs sisters,
  while the equivalence for a male consists of only
  that male.

49
Equivalence Relations

• There is a correspondence between equivalence in
  relations and partitions in sets.
• ?b,c? ? sameEye ? b and c have same eye colour

50
Equivalence Relations

• Theorem 1 Let S be a partition of a set X.
  Define (x R y) to mean that for some S in S,
  both x and y belong to S. Then R is reflexive,
  symmetric, and transitive.
• Stating this another way, a relation that is
  reflexive, symmetric, and transitive is an
  equivalence relation.

51
Equivalence Relation()

• Let S be a partition of X which gives rise to an
  equivalence relation R on X.
• X 1,2,3,4,5,6
• S 1,3,5,4,2,6 partition X
• The relation R on X is an equivalence relation
  therefore R is reflexive, symmetric, and
  transitive. Draw S. List the ordered pairs of the
  relation R on X. Draw the diagraph of R.

52
Equivalence Relation()

• Draw S 1,3,5,4,2,6

11
22
.
.
2
62
1
.
.
6
3
31
.
5
51
.
4
43
Venn Diagram The equivalence classes are
subscripted according to their respective
partitions
53
Equivalence Relation()

• List the ordered pairs of the relation R on X.
• R (1,1), (1,3),(1,5),(2,2),(2,6),
• (3,1),(3,3),(3,5),(4,4),(5,1),
• (5,3),(5,5),(6,2),(6,6)

54
Graph of Equivalence Relation

• R (1,1), (1,3),(1,5),(2,2),(2,6),
• (3,1),(3,3),(3,5),(4,4),(5,1),
• (5,3),(5,5),(6,2),(6,6)

1
2
3
5
4
6
55
Order Relations

• The order relation compares some members of a
  set. A typical order relation is lt on integers.
• An order relation need not be a comparison of
  every member of a set, e.g. while A and B may be
  parents parent(A,B) may not hold.

56
Equivalence Relation()

• Draw P a,c,e,d,b,f

a1
b2
.
.
b
f2
a
.
.
f
c
c1
.
e
e1
.
d
d3
Venn Diagram The equivalence classes are
subscripted according to their respective
partitions
57
Equivalence Relation()

• Draw the diagraph of R.

b
a
c
e
d
f
58
Equivalence Relation()

• List the ordered pairs of the relation R on X.
• R (a,a), (a,c),(a,e),(b,b),(b,e),
• (c,a),(c,c),(c,e),(d,d),(e,a),
• (e,c),(e,e),(f,b),(f,f)

59
Def. Partial Order Relation

• A binary relation ? on a set b is called a
  partial order on B if it is Reflexive,
  Antisymmetric, and Transitive. ltB, ? gt
  is called a partially ordered set or poset.
• We will write partial orders using a?b and b?c.
  This notation differs from the course text book.

60
Partial Order Relation1
Is this relation reflexive, antisymmetric, and
transitive?
61
Hasse Diagrams

• Hasse diagrams are a form of simplified digraph
  that allow us to represent partial orders. They
  are simplified as follows.
• Loops may be suppressed
• Arrows not necessary, read from bottom to top.
• Transitive arrows may be suppressed.

62
Order Relation on Power set.

• lt?,?gt is a poset and ? is a partial order on ?.
• Let B be a set. Then lt?B,?gt is a poset and ? is
  a partial order on ?B (the power set of B), since
  is symmetric, antisymmetric, and transitive.
• Let C be the set of courses offered at DIT.
  Define the relation ? by c1?c2 if c1c2 or c1 is
  a prerequisite of c2. Then ltC,?gt is a poset and ?
  is a partial order on C.

63
Order Relation on Power set1.

• A partially order set can be represented with
  using a POSET diagram. The POSET diagram on the
  right is based on the power set (all possible
  subsets) of the three element set a, b, c.
  These subsets form a special kind of partial
  order that is referred to as a lattice

64
Order Relation.

• The POSET diagram on the right is based on the
  power set (all possible subsets) of the three
  element set 1, 2, 3.

65
Partial Order Relation on Divisibility1.

• The set
• A 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
• of all divisors of 60, partially ordered by
  divisibility.

66
Order Relations

• In the construction of a house, certain jobs have
  to be done before other jobs. Let J be the set of
  jobs and let defined on J be defined by j1 ? j2
  if j1has to be completed before j2 can be
  started. ltJ, ?gt is a poset.

67
Total Order Relations

• With partial orders because not all elements need
  to be comparable. But some total relations give
  rise to total order. For example less than or
  equal to" is a total relation over the set of
  real numbers, because for two numbers either the
  first is less than or equal to the second, or the
  second is less than or equal to the first. We can
  define a total order in two ways 1)in terms of
  membership and 2)in terms of operations on sets.

68
Total Order Definition

• A partial order ? over B is called a total order
  or linear order if
• Def 1
• (?b,c b?c ? c?b)
• Def 2
• iff ???-1 B ? B
• ltB,?gt is called a linearly ordered set or a
  chain.

69
Total Orders

• lt?,?gt and lt?,?gt are chains
• Let set S contain more than one element. Then ?
  over ?S is not a total order. Because if b and c
  are distinct elements of S, then neither b?c
  nor c?b holds.
• Any subset of a totally ordered set, with the
  restriction of the order on the whole set.
• Any partially ordered set X where every two
  elements are comparable (i.e. if a,b are members
  of X either ab or ba or both).

70
Partial but not Total Order

• Let C be the set of courses at DIT. Let b ?
  c mean that b c or b is a prerequisite for c.
  The relation is a partial order but not a total
  order.

71
Total Orders

• The letters of the alphabet ordered by the
  standard dictionary order (e.g., A lt B lt C) is a
  partial order.

72
UML Association as a Relation

• A representation of a UML association as 1)a
  diagram and 2)a relation

73
UML Aggregation Relation

• Informal whole-part relationships can be
  modelled using aggregation
• a specialized form of an association
• can have standard annotations on ends

74
UML Cyclic Object Structures

• Aggregation is useful for ruling out invalid
  cyclic object structures
• eg where an assembly contains itself, directly or
  indirectly

75
UML Properties of Aggregation

• Aggregation rules this out because it is
• antisymmetric an object cant link to itself
• transitive if a links to b and b to c, a links
  to c

part end
whole end
76
Meaning of Aggregation

• Sometimes there is a conflict
• E.g. people cannot be their own ancestors
• this can be specified using aggregation
• but a persons ancestors are not a part of them!
• The aggregation does not make sense in this case

77
Meaning (semantics) of Aggregation Relation

• A hand may be considered as part of a person and
  a person may be considered as part of an
  orchestra, but we would not consider hand as part
  of an orchestra.
• Heuristic If the part moves, can one deduce that
  the whole moves with it in normal circumstances.