unit unit 04 relations it disiciplineitd1111 discrete mathematics & statistics stdtlp1 unit 4...

IT DisiciplineIT Disicipline ITD1111 Discrete Mathematics & Statistics STDTLPITD1111 Discrete Mathematics & Statistics STDTLP 11

UnitUnit 04 Relations

Unit 4 Relations



4.1 Introduction to Relations

Relationships between elements of sets

occur in many contexts. We have many

examples in everyday life such as those

between companies, schools and their

telephone numbers, a person and a relative,

a student and his/her student number.



Introduction to Relations

Relations can be used to solve problems such

as determining which pairs of cities are linked

by airline flights in a network, or producing a

useful way to store information in computer

databases.




Definition: Let A and B be sets. The

Cartesian product of A and B, is defined by

E.g. If A={1,2} and B={a,b,c}, then

B}b and Aa :b){(a, BA c)}(2,b),(2,a),(2,c),(1,b),(1,a),{(1, BA




Relationships between elements of sets are

represented using a structure called relation.

Definition: Let A and B be sets. A relation R

from A to B (a binary relation) is a subset of BA



4.2 Relations and Their Properties

Use ordered pairs (a, b) to represent the relationship between elements of two sets.

Example 4.2-1

Let A be the set of students in the ICT department,

Let B be the set of courses,

Let R be the relation that consists of those pairs (a, b) where a is a student enrolled in course b.

Then we may have

(Raymond, 41300), (John, 41983) belonging to R.



Example 4.2-2

Let A={0,1,2} and B={a,b}.

If R={(0,a), (0,b), (1,a), (2,b)}, then

0 is related to a

but 1 is not related to b.



Relations can be represented graphically and in tabular form

Graphical method Tabular form

R={(0,a), (0,b), (1,a), (2,b)}

0

1

2

a

b

R a b

0 X X

1 X

2 X



4.3 Relations on a Set

Relations from a set A to itself are of special interest. • Definition: A relation on a set A is a relation from

A to A.

Example 4.3-1

Let A = {1, 2, 3, 4 }. Which ordered pairs are in

the relation ?

R ={(a, b) AA : a divides b } ?R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4,4)}



Example 4.3-2Which of the following relations contain the ordered pairs (1, 1), (1, 2), (2, 1), (1, -1) or (2, 2) ?

SolutionR1 = {(a, b) : a < b } {(1, 1), (1, 2), (2, 2)}

R2 = {(a, b) : a > b } {(2, 1), (1, -1)}

R3 = {(a, b) : a = b or a = -b} {(1, 1), (1, -1), (2, 2)}

R4 = {(a, b) : a = b } {(1, 1), (2, 2)}

R5 = {(a, b) : a = b + 1 } {(2, 1)}

R6 = {(a, b) : a + b < 3 }. {(1, 1), (1, 2), (2, 1), (1,-1)}



4.4 Combining Relations

Two relations from A to B can be combined

using the set operations of union ,

intersection and difference \. Consider

the following examples.



Example 4.4-1

Let R1 = {(1, 1), (2, 2), (3, 3)} and

R2 = {(1, 1), (1, 2), (1, 3), (1,4)}

then :

R1 R2 = {(1, 1), (1, 2), (1, 3), (1,4), (2, 2), (3, 3)}

R1 R2 = {(1, 1)}

R1 \ R2 = {(2, 2), (3, 3)}

R2 \ R1 = {(1, 2), (1, 3), (1,4)}



4.5 Representing Relations

– List its ordered pairs

– Graphical method

– Tabular form

– Use zero-one matrices

– Use directed graphs



4.5.1 Representing relations using matricesExample 4.5-1Suppose that A = {1, 2, 3} and B ={1, 2}. Let R be the relation from A to B such that it contains (a, b) if

a A, b B, and a > b.What is the matrix representing R ?Since R = {(2, 1), (3, 1), (3, 2)}, the matrix for R is :

11

01

00

RM



4.5.2 Representing relations using directed graphs

Example 4.5-2

R = {(1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,3),

(4,1), (4,3)}

1

3

2

4



4.6 Properties of Binary Relations

The most direct way to express a relationship

between two sets was to use ordered pairs. For

this reason, sets of ordered pairs are called

binary relations.



4.6.1 Reflexive Property of a Binary Relation

Definition:

A relation R on a set A is called reflexive if (a,

a) R for every element a A.



R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}

Not reflexive because 3 A but (3,3) R1

R2 = {(1, 1), (1, 2), (2, 1)}Not reflexive because, say, 4 A but (4, 4) R2

R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}

Reflexive

Example 4.6-1 Consider the following relations on A={1,2,3,4}. Which of these relations are reflexive?



R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}

Not reflexive - (1, 1) ?

R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3),

(2, 4),(3, 3), (3, 4), (4, 4)}

Reflexive - Why ?

R6 = {(3, 4)}

Not Reflexive - Why ?



4.6.2 Symmetric Property of a Binary Relation

A relation R on a set A is called symmetric if

for all a, b A, (a, b) R implies (b, a) R .

Definitions:

A relation R on a set A is called antisymmetric

if for all a, b A,

(a, b) R and (b, a) R implies a = b.



R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}

Not symmetric - (3, 4) but there is no (4, 3)

Not antisymmetric - (1, 2) & (2, 1) but 12

R2 = {(1, 1), (1, 2), (2, 1)}

Symmetric


R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}

Symmetric


Example 4.6-2 Which of the relations are symmetric and which are antisymmetric?



R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}Not symmetric - (2, 1) but no (1, 2)Antisymmetric

R5 = { (1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4),

(3, 3), (3, 4), (4, 4) }Not symmetric - (1, 3) but no (3, 1)Antisymmetric

R6 = {(3, 4)}

Not symmetric - (3, 4) but no (4, 3)Antisymmetric



4.6.3 Transitive Property of a Binary RelationDefinition:A relation R on a set A is called transitive if whenever (a, b) R and (b, c) R then(a, c) R, for a, b, c A.

Example 4.6-3 Which of the following relations are transitive?R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}

Not transitive because- (3, 4) & (4, 1) R1 but (3, 1) R1



R2 = {(1, 1), (1, 2), (2, 1)}Not transitive because

- (2, 1) & (1, 2) R2 but (2, 2) R2

R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}Not transitive - (4, 1) & (1, 2) R3 but (4, 2) R3

R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}Transitive

R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}

Transitive

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