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

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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.
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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
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