Discrete Mathematics Introduction

Discrete Math T. Serino Discrete Math is not a subject that can be easily defined. It was originally created for computer science students working on algorithms and recursion, but in its maturity, has become much more available and real to people of many fields and academic levels. If I had to define Discrete Mathematics, I would have to define is as the study of many mathematical topics in which data that typically involve counts (such as number of people or objects) is organized in order to solve a wide array of applications.

Discrete Math T. Serino APPLICATIONS ALGORITHMS AND REASONING COUNTING PROBABILITY GRAPH THEORY Social Choice MATRIX ALGEBRA RECURSION ITERATION This diagram gives a nice picture of the topics encompassed by Discrete Mathematics.

Discrete Math Logic Set Theory Social Choice Flow Charts GRAPH THEORY T. Serino APPLICATIONS ALGORITHMS AND REASONING COUNTING PROBABILITY GRAPH THEORY Social Choice MATRIX ALGEBRA RECURSION ITERATION ALGORITHMS AND REASONING Logic Set Theory Flow Charts

Discrete Math Fair Division Social Social Choice Choice Voting Methods T. Serino APPLICATIONS ALGORITHMS AND REASONING COUNTING PROBABILITY GRAPH THEORY Social Choice MATRIX ALGEBRA RECURSION ITERATION Social Choice Fair Division Voting Methods Apportionment

Discrete Math Circuits and Paths Social Choice Map Coloring T. Serino APPLICATIONS ALGORITHMS AND REASONING COUNTING PROBABILITY GRAPH THEORY Social Choice MATRIX ALGEBRA RECURSION ITERATION GRAPH THEORY Circuits and Paths Map Coloring Venn Diagrams

Discrete Math Fractal Geometry Chaos Theory Social Choice T. Serino APPLICATIONS ALGORITHMS AND REASONING COUNTING PROBABILITY GRAPH THEORY Social Choice MATRIX ALGEBRA RECURSION ITERATION Fractal Geometry Chaos Theory RECURSION ITERATION Data Compression

Geometric Transformations Discrete Math T. Serino APPLICATIONS ALGORITHMS AND REASONING COUNTING PROBABILITY GRAPH THEORY Social Choice MATRIX ALGEBRA RECURSION ITERATION Dilation Geometric Transformations MATRIX ALGEBRA Translations

Discrete Math Combinatorics Social Choice Expected Values Permutations T. Serino APPLICATIONS ALGORITHMS AND REASONING COUNTING PROBABILITY GRAPH THEORY Social Choice MATRIX ALGEBRA RECURSION ITERATION Combinatorics Expected Values Permutations COUNTING AND PROBABILITY

What's the fastest route? Discrete Math T. Serino Optimization APPLICATIONS ALGORITHMS AND REASONING COUNTING PROBABILITY GRAPH THEORY Social Choice MATRIX ALGEBRA RECURSION ITERATION Who's the best? What is fair? APPLICATIONS What's the fastest route? What should we expect?

(The height of a flower) Discrete Math T. Serino One good way to understand discrete data is to compare it to continuous data. Discrete Data vs. Continuous Not countable, always changing. Countable, separate events. (The height of a flower) (Number of students in class each day)

Discrete Math Discrete Mathematics Continuous Not countable, T. Serino Discrete Mathematics vs. Continuous Not countable, always changing. Countable, separate events. vs. Amount of water being poured into a glass. Number of Ice cubes being Stacked in a glass. vs. The age of a student. The number of Birthdays a student has had. vs. A child’s height from The ground while Gliding down a slide. A person’s height from the ground as he/she reaches each step on a stairway. vs.

Sample problem T. Serino You and a friend have a large heart-shaped chocolate bar which you want to split equally. One possible fair way to do this leaving you both happy is for one of you to split it and the other one to choose the first piece. (Think about why this can be considered fair.) What is a fair way for three people to split the candy bar into three pieces so that each of the three people feels that they have a fair share?

Sample Problem T. Serino Trucks belonging to companies such as UPS and FedEx have packages to deliver to a list of destinations, generally between 100 and 200, destinations per truck per day. Travel time between any two destinations is known or can be estimated. In what order should the driver deliver the packages in order to finish in the least amount of time or using the least amount of gas?

Sample Problem T. Serino Selecting the city in which the next Olympic Games are to be held is a decision entrusted to the members of the International Olympic Committee. It is a decision that has a tremendous economic and political impact on the cities involved and it always generates controversy. On September 23, 1993, the 89 members of the International Olympic Committee met in Monaco to vote on the site for the 2000 Summer Olympics. Five cities made bids: Beijing, Berlin, Istanbul, Manchester and Sydney. We perhaps are most familiar with the plurality method of voting: the city with the most first place votes wins. But is this the fairest method? What other voting systems are there and what voting system was used to determine the winning city in this case?

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