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MATH 245 CH2

# MATH 245 CH2 - DISCRETE MATHEMATICS W W L CHEN c W W L Chen...

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DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 2 RELATIONS AND FUNCTIONS 2.1. Relations We start by considering a simple example. Let S denote the set of all students at Beachbum University, and let T denote the set of all teaching staff there. For every student s S and every teaching staff t T , exactly one of the following is true: s has attended a lecture given by t ; or s has not attended a lecture given by t . We now define a relation R as follows. Let s S and t T . We say that s R t if s has attended a lecture given by t . If we now look at all possible pairs ( s, t ) of students and teaching staff, then some of these pairs will satisfy the relation while other pairs may not. To put it in a slightly different way, we can say that the relation R can be represented by the collection of all pairs ( s, t ) where s R t . This is a subcollection of the set of all possible pairs ( s, t ). Definition. Let A and B be sets. The set A × B = { ( a, b ) : a A and b B } is called the cartesian product of the sets A and B . In other words, A × B is the set of all ordered pairs ( a, b ), where a A and b B . Definition. Let A and B be sets. By a relation R on A and B , we mean a subset of the cartesian product A × B . Remark. There are many instances when A = B . Then by a relation R on A , we mean a subset of the cartesian product A × A . Example 2.1.1. Let A = { 1 , 2 , 3 , 4 } . Define a relation R on A by writing ( x, y ) ∈ R if x < y . Then R = { (1 , 2) , (1 , 3) , (1 , 4) , (2 , 3) , (2 , 4) , (3 , 4) } . Chapter 2 : Relations and Functions page 1 of 8

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Discrete Mathematics c W W L Chen, 1982, 2008 Example 2.1.2. Let A be the power set of the set { 1 , 2 } ; in other words, A = {∅ , { 1 } , { 2 } , { 1 , 2 }} is the set of subsets of the set { 1 , 2 } . Then it is not too difficult to see that R = { ( , { 1 } ) , ( , { 2 } ) , ( , { 1 , 2 } ) , ( { 1 } , { 1 , 2 } ) , ( { 2 } , { 1 , 2 } ) } is a relation on A where ( P, Q ) ∈ R if P Q . Example 2.1.3. Of course, nobody attends any lectures at Beachbum University, so that clearly R = ! 2.2. Equivalence Relations We begin by considering a familiar example. Example 2.2.1. A rational number is a number of the form p/q , where p Z and q N . This can also be viewed as an ordered pair ( p, q ), where p Z and q N . Let A = { ( p, q ) : p Z and q N } . We can define a relation R on A by writing (( p 1 , q 1 ) , ( p 2 , q 2 )) ∈ R if p 1 q 2 = p 2 q 1 , i.e. if p 1 /q 1 = p 2 /q 2 . This relation R has some rather interesting properties: (( p, q ) , ( p, q )) ∈ R for all ( p, q ) A ; whenever (( p 1 , q 1 ) , ( p 2 , q 2 )) ∈ R , we have (( p 2 , q 2 ) , ( p 1 , q 1 )) ∈ R ; and whenever (( p 1 , q 1 ) , ( p 2 , q 2 )) ∈ R and (( p 2 , q 2 ) , ( p 3 , q 3 )) ∈ R , we have (( p 1 , q 1 ) , ( p 3 , q 3 )) ∈ R .
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