Representing Relations

1
Representing Relations

• Rosen, section 7.3
• CS/APMA 202
• Aaron Bloomfield

2
In this slide set

• Matrix review
• Two ways to represent relations
• Via matrices
• Via directed graphs

3
Matrix review

• This is from Rosen, page 201 and 202
• We will only be dealing with zero-one matrices
• Each element in the matrix is either a 0 or a 1
• These matrices will be used for Boolean
  operations
• 1 is true, 0 is false

4
Matrix transposition

• Given a matrix M, the transposition of M, denoted
  Mt, is the matrix obtained by switching the
  columns and rows of M
• In a square matrix, the main diagonal stays
  unchanged

5
Matrix join

• A join of two matrices performs a Boolean OR on
  each relative entry of the matrices
• Matrices must be the same size
• Denoted by the or symbol ?

6
Matrix meet

• A meet of two matrices performs a Boolean AND on
  each relative entry of the matrices
• Matrices must be the same size
• Denoted by the or symbol ?

7
Matrix Boolean product

• A Boolean product of two matrices is similar to
  matrix multiplication
• Instead of the sum of the products, its the
  conjunction (and) of the disjunctions (ors)
• Denoted by the or symbol ?

8
Relations using matrices

• List the elements of sets A and B in a particular
  order
• Order doesnt matter, but well generally use
  ascending order
• Create a matrix

9
Relations using matrices

• Consider the relation of who is enrolled in which
  class
• Let A Alice, Bob, Claire, Dan
• Let B CS101, CS201, CS202
• R (a,b) person a is enrolled in course b

CS101 CS201 CS202
Alice X
Bob X X
Claire
Dan X X
10
Relations using matrices

• What is it good for?
• It is how computers view relations
• A 2-dimensional array
• Very easy to view relationship properties
• We will generally consider relations on a single
  set
• In other words, the domain and co-domain are the
  same set
• And the matrix is square

11
Reflexivity

• Consider a reflexive relation
• One which every element is related to itself
• Let A 1, 2, 3, 4, 5

If the center (main) diagonal is all 1s, a
relation is reflexive
12
Irreflexivity

• Consider a reflexive relation lt
• One which every element is not related to itself
• Let A 1, 2, 3, 4, 5

If the center (main) diagonal is all 0s, a
relation is irreflexive
13
Symmetry

• Consider an symmetric relation R
• One which if a is related to b then b is related
  to a for all (a,b)
• Let A 1, 2, 3, 4, 5

• If, for every value, it is the equal to the value
  in its transposed position, then the relation is
  symmetric

14
Asymmetry

• Consider an asymmetric relation lt
• One which if a is related to b then b is not
  related to a for all (a,b)
• Let A 1, 2, 3, 4, 5

• If, for every value and the value in its
  transposed position, if they are not both 1, then
  the relation is asymmetric
• An asymmetric relation must also be irreflexive
• Thus, the main diagonal must be all 0s

15
Antisymmetry

• Consider an antisymmetric relation
• One which if a is related to b then b is not
  related to a unless ab for all (a,b)
• Let A 1, 2, 3, 4, 5

• If, for every value and the value in its
  transposed position, if they are not both 1, then
  the relation is antisymmetric
• The center diagonal can have both 1s and 0s

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Transitivity

• Consider an transitive relation
• One which if a is related to b and b is related
  to c then a is related to c for all (a,b), (b,c)
  and (a,c)
• Let A 1, 2, 3, 4, 5

• If, for every spot (a,b) and (b,c) that each have
  a 1, there is a 1 at (a,c), then the relation is
  transitive
• Matrices dont show this property easily

17
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18
End of lecture on 21 April 2005

• I think!

19
Combining relations via Boolean operators

• Example 4 from Rosen, section 7.3
• Let
• Join
• Meet

20
Combining relations via relation composition

• Example 4 from Rosen, section 7.3
• Let
• But why is this the case?

d e f
g h i
a b c
d e f
g h i
a b c
21
Representing relations using directed graphs

• A directed graph consists of
• A set V of vertices (or nodes)
• A set E of edges (or arcs)
• If (a, b) is in the relation, then there is an
  arrow from a to b
• Will generally use relations on a single set
• Consider our relation R (a,b) a divides b
• Old way

22
Reflexivity

• Consider a reflexive relation
• One which every element is related to itself
• Let A 1, 2, 3, 4, 5

If every node has a loop, a relation is reflexive
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Irreflexivity

• Consider a reflexive relation lt
• One which every element is not related to itself
• Let A 1, 2, 3, 4, 5

If every node does not have a loop, a relation is
irreflexive
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Symmetry

• Consider an symmetric relation R
• One which if a is related to b then b is related
  to a for all (a,b)
• Let A 1, 2, 3, 4, 5

• If, for every edge, there is an edge in the other
  direction, then the relation is symmetric
• Loops are allowed, and do not need edges in the
  other direction

Called anti- parallel pairs
Note that this relation is neither reflexive nor
irreflexive!
25
Asymmetry

• Consider an asymmetric relation lt
• One which if a is related to b then b is not
  related to a for all (a,b)
• Let A 1, 2, 3, 4, 5

• A digraph is asymmetric if
• If, for every edge, there is not an edge in the
  other direction, then the relation is asymmetric
• Loops are not allowed in an asymmetric digraph
  (recall it must be irreflexive)

26
Antisymmetry

• Consider an antisymmetric relation
• One which if a is related to b then b is not
  related to a unless ab for all (a,b)
• Let A 1, 2, 3, 4, 5

• If, for every edge, there is not an edge in the
  other direction, then the relation is
  antisymmetric
• Loops are allowed in the digraph

27
Transitivity

• Consider an transitive relation
• One which if a is related to b and b is related
  to c then a is related to c for all (a,b), (b,c)
  and (a,c)
• Let A 1, 2, 3, 4, 5

• A digraph is transitive if, for there is a edge
  from a to c when there is a edge from a to b and
  from b to c

28
Applications of digraphs MapQuest

• Not reflexive
• Is irreflexive
• Not symmetric
• Not asymmetric
• Not antisymmetric
• Not transitive

• Not reflexive
• Is irreflexive
• Is symmetric
• Not asymmetric
• Not antisymmetric
• Not transitive

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Rosen, questions 31 32, section 7.3
Which of the graphs are reflexive, irreflexive,
symmetric, asymmetric, antisymmetric, or
transitive
23 24 25 26 27 28
Reflexive Y Y Y
Irreflexive Y Y
Symmetric Y Y
Asymmetric Y
Anti-symmetric Y Y
Transitive Y
30
Rosen, section 7.1 (sic) question 45 (a)

• How many symmetric relations are there on a set
  with n elements?
• Solution guide explanation is pretty poorly
  worded
• So instead well use matrices

31
Rosen, section 7.1 (sic) question 45 (a)

• Consider the matrix representing symmetric
  relation R on a set with n elements
• The center diagonal can have any values
• Once the upper triangle is determined, the
  lower triangle must be the transposed version
  of the upper one
• How many ways are there to fill in the center
  diagonal and the upper triangle?
• There are n2 elements in the matrix
• There are n elements in the center diagonal
• Thus, there are 2n ways to fill in 0s and 1s in
  the diagonal
• Thus, there are (n2-n)/2 elements in each
  triangle
• Thus, there are ways to fill
  in 0s and 1s in the triangle
• Answer there are
  possible symmetric relations on a set
  with n elements

32
Quick survey

• I felt I understood the material in this slide
  set
• Very well
• With some review, Ill be good
• Not really
• Not at all

33
Quick survey

• The pace of the lecture for this slide set was
• Fast
• About right
• A little slow
• Too slow

34
Quick survey

• How interesting was the material in this slide
  set? Be honest!
• Wow! That was SOOOOOO cool!
• Somewhat interesting
• Rather borting
• Zzzzzzzzzzz

35
Biggest software errors

• Ariane 5 rocket explosion (1996)
• Due to loss of precision converting 64-bit double
  to 16-bit int
• Pentium division error (1994)
• Due to incomplete look-up table (like an array)
• Patriot-Scud missile error (1991)
• Rounding error on the time
• The missile did not intercept an incoming Scud
  missile, leaving 28 dead and 98 wounded
• Mars Climate Orbiter (1999)
• Onboard used metric units ground computer used
  English units
• ATT long distance (1990)
• Wrong break statement in C code
• Therac-25, X-ray (1975-1987)
• Badly designed software led to radiation overdose
  in chemotherapy patients
• NE US power blackout (2003)
• Flaw in GE software contributed to it
• References http//www5.in.tum.de/huckle/bugse.ht
  ml, http//en.wikipedia.org/wiki/Computer_bug,
  http//www.cs.tau.ac.il/nachumd/verify/horror.htm
  l