IT Disicipline IT Disicipline ITD1111 Discrete Mathematics & Statistics ITD1111 Discrete Mathematics & Statistics STDTLP STDTLP 1 Unit Unit 04 Relations Unit 4 Relations
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UnitUnit 04 Relations
Unit 4 Relations
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UnitUnit 04 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.
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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.
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UnitUnit 04 Relations
Introduction to Relations
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
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UnitUnit 04 Relations
Introduction to Relations
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
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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.
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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.
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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
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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)}
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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)}
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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.
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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)}
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4.5 Representing Relations
– List its ordered pairs
– Graphical method
– Tabular form
– Use zero-one matrices
– Use directed graphs
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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
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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
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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.
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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.
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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?
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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 ?
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UnitUnit 04 Relations
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.
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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
Not antisymmetric - (1, 2) & (2, 1) but 12
R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
Symmetric
Not antisymmetric - (1, 4) & (4, 1) but 14
Example 4.6-2 Which of the relations are symmetric and which are antisymmetric?
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UnitUnit 04 Relations
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
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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
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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