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Representing Relations

Epp section ???

CS 202Aaron Bloomfield

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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 Booleanoperations 1 is true, 0 is false

0101

0101

0010

0001

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Matrix transposition

Given a matrix M, the transposition ofM, denoted Mt, isthe matrix obtained by switching the columns and rows ofM

In a square matrix, the maindiagonal stays unchanged

=

654321M

=

16151413

1211109

8765

4321

M

=

63

52

41t

M

=

161284

151173

14106213951

tM

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Matrix join

Ajoin 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:

=

0111

1101

01100111

0011

1100

01100110

0101

0101

00100001

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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:

=

0001

0100

00100000

0011

1100

01100110

0101

0101

00100001

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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, its the conjunction (and) ofthe 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 =

=

1110

1110

0110

0110

0011

1100

0110

0110

0101

0101

0010

0001

=

0

0011

1100

0110

0110

0101

0101

0010

0001

=

0011

1100

0110

0110

0101

0101

0010

0001

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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 }

=

110

000

110

001

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 }

=

10000

11000

11100

11110

11111

M

If the center (main)

diagonal is all 1s, a

relation is reflexive

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Irreflexivity

Consider a reflexive relation: