{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# ln8 - Massachusetts Institute of Technology 6.042J/18.062J...

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

Massachusetts Institute of Technology Course Notes 8 6.042J/18.062J, Fall ’02 : Mathematics for Computer Science October 21 Professor Albert Meyer and Dr. Radhika Nagpal revised October 24, 2002, 874 minutes Basic Counting, Pigeonholing, Permutations 1 Counting by Matching Counting is a theme throughout discrete mathematics: how many leaves in a tree, minimal color- ings of a graph, trees with a given set of vertices, five-card hands in a deck of fifty-two, consistent rankings of players in a tournament, stable marriages given boy’s and girl’s preferences, and so on. A good way to count things is to match up things to be counted with other things that we know how to count. We saw an example of this early in the term when we counted the size of a powerset of a set of size n by finding an exact matching between elements of the powerset and the 2 n binary strings of length n . The matching doesn’t have to be exact, i.e. , a bijection, to be informative. For example, suppose we want to determine the cardinality of the set of watches in the 6.042 classroom on a typical day. The set of watches can be correlated with the set of people in the room; specifically, for each person there is at most one watch (at least, let’s assume this). Now we know something about the cardinality of the set of students, since there are only 146 people signed up for 6.042. There are also three lecturers and eight TA’s, and these would typically be the only nonstudents in the room. So we can conclude that there are at most 157 watches in the classroom on a typical day. This type of argument is very simple, but also quite powerful. We will see how to use such simple arguments to prove results that are hard to obtain any other way. 2 Matchings as Bijections The “matching up” we talked about more precisely refers to finding injections, surjections, and bijections between things we want to count and things to be counted. The following Theorem formally justifies this kind of counting: Theorem 2.1. Let A and B be finite sets and f : from A to B be a function. If 1. f is a bijection, then | A | = | B | , 2. f is an injection, then | A | ≤ | B | , 3. f is a surjection, then | A | ≥ | B | . Copyright © 2002, Prof. Albert R. Meyer . All rights reserved.

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

View Full Document
2 Course Notes 8: Basic Counting, Pigeonholing, Permutations This is one of those theorems that is so fundamental that it’s not clear what simpler axioms are appropriate to use in proving it. In fact, we can’t prove it yet, because we haven’t defined the concept that it’s all about, namely, the size or cardinality , | A | , of a finite set, A . Intuitively, a set, A , has n elements if it equals { a 1 , a 2 , . . . , a n } where the a i are all different. Now ellipsis is dangerous, so we should avoid it in a definition this basic. What is the notation “ a 1 , a 2 , . . . , a n ” intended to convey? It means that there is a first element, a 1 , and a second element, a 2 , and in general, given any i n , there is an i th element a i . Also, all the a i ’s for different i ’s are different. This explains how we arrive at a rigorous definition: Definition 2.2. A set A has cardinality n N
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern