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Representing Relations

Based on Aaron BloomfieldModified by Longin Jan Latecki

Rosen, Section 8.3

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In this slide set…

• Matrix review• Two ways to represent relations

– Via matrices– Via directed graphs

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Matrix review

• 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

0101010100100001

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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

654321

M

16151413121110987654321

M

635241

tM

16128415117314106213951

tM

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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:

0111110101100111

0011110001100110

0101010100100001

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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:

0001010000100000

0011110001100110

0101010100100001

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Matrix Boolean product• A Boolean product of two matrices is similar to matrix

multiplication

– Instead of the sum of the products, it’s the conjunction (and) of the disjunctions (ors)

– Denoted by the or symbol:

1,44,11,33,11,22,11,11,11,1 **** babababac

1,44,11,33,11,22,11,11,11,1 babababac

1110111001100110

0011110001100110

0101010100100001

0

0011110001100110

0101010100100001

0011110001100110

0101010100100001

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Relations using matrices

• List the elements of sets A and B in a particular order– Order doesn’t matter, but we’ll generally use

ascending order• Create a matrix ][ ijR mM

Rba

Rbam

ji

jiij ),( if 0

),( if 1

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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 }

110000110001

RM

CS101 CS201 CS202

Alice X

Bob X X

Claire

Dan X X

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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

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Reflexivity

• Consider a reflexive relation: ≤– One which every element is related to itself– Let A = { 1, 2, 3, 4, 5 }

1000011000111001111011111

M

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

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Irreflexivity

• Consider a reflexive relation: <– One which every element is not related to itself– Let A = { 1, 2, 3, 4, 5 }

0000010000110001110011110

M

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

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1001001001000011000001101

M

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

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0000010000110001110011110

M

Asymmetry

• Consider an asymmetric relation: <– 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 0’s

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1000011000111001111011111

M

Antisymmetry

• Consider an antisymmetric relation: ≤– One which if a is related to b then b is not related to a

unless a=b 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 1’s and 0’s

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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 }

1000011000111001111011111

M

• 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 don’t show this property easily

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Combining relations: via Boolean operators

• Example 4 from Rosen, section 8.3

• Let:

• Join:

• Meet:

010001101

RM

001110101

SM

011111101

SRSR MMM

000000101

SRSR MMM

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Combining relations: via relation composition

• Example 4 from Rosen, section 8.3

• Let:

• But why is this the case?

010001101

RM

001110101

SM

110101101

SRRS MMM

a

b

c

d

e

f

d e f g h i

a

b

c

g h i

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Find the matrix representing the relations S ° R:

SRRS MMM

R Sa

b

c

d

f

e

g

i

h

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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:1

2

3

4

1

2

3

4

1 2

3 4

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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

1 2

5 3

4

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Irreflexivity

• Consider a reflexive relation: <– 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

1 2

5 3

4

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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

1 2

5 3

4 Note that this relation is neither reflexive nor irreflexive!

Called anti-parallel pairs

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Asymmetry

• Consider an asymmetric relation: <– 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:1. If, for every edge, there is not

an edge in the other direction, then the relation is asymmetric

2. Loops are not allowed in an asymmetric digraph (recall it must be irreflexive)

1 2

5 3

4

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Antisymmetry

• Consider an antisymmetric relation: ≤– One which if a is related to b then b is not related to a

unless a=b for all (a,b)– Let A = { 1, 2, 3, 4, 5 }

1 2

5 3

4

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

• Loops are allowed in the digraph

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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 }

1 2

5 3

4

• 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

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Applications of digraphs: MapQuest

Start

End

•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 8.3

Which of the graphs are reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive

23 24 25 26 27 28Reflexive Y Y YIrreflexive Y YSymmetric Y YAsymmetric YAnti-symmetric

Y Y

Transitive Y Y

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Rosen, Section 8.1 question 45(a)

• How many symmetric relations are there on a set with n elements?

• Solution guide explanation is pretty poorly worded

• So instead we’ll use matrices

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Rosen, Section 8.1 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 0’s and 1’s in the diagonal• Thus, there are (n2-n)/2 elements in each triangle

– Thus, there are ways to fill in 0’s and 1’s in the triangle

• Answer: there are possible symmetric relations on a set with n elements

10100

01101

2/)( 2

2 nn

2/)(2/)( 22

22*2 nnnnn

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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• AT&T 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.html, http://en.wikipedia.org/wiki/Computer_bug, http://www.cs.tau.ac.il/~nachumd/verify/horror.html

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