{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 .
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern