Mathematical logic, set theory, the real numbers, induction and recursion, summation notation, asymptotic notation, number theory, relations, graphs, counting, linear algebra, finite fields. Along the way, we recall some notions from discrete mathematics that you are. This course acts as an introduction to contemporary discrete mathematics. Discrete mathematics counting saad mneimneh 1 n choose k consider the problem of seating n people on n chairs. Counting principle algebra 2, discrete mathematics and. From a set s x, y, z by taking two at a time, all permutations are. Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. Discrete mathematics and its applications lecture 3. Chapter 4 counting principles learning objectives on completing this chapter, you should be able to.
For a pair of sets a and b, a b denotes theircartesian product. Principle of inclusionexclusion how many bit strings of length eight either start with 1 or end with 00. One way to formally introduce counting principles is through countable sets. Discrete mathematics deals with objects that come in discrete bundles, e.
The rule of sum addition principle and the rule of product multiplication principle are stated as below. We must count objects to solve many different types of problems, like the determining whether there are enough telephone numbers or internet protocal ip addresses to meet demand. Discrete mathematics rule of sum and rule of product. Ma8351 notes discrete mathematics to extend students logical and mathematical maturity and ability to deal with abstraction. Chapter 4 counting principles discrete mathematics and. Discrete mathematics notes ma8351 pdf free download. Home highlights for high school mathematics combinatorics. Here are two more, straightforward examples where the question is to count the steps of. The basics of counting the pigeonhole principle permutations and combinations binomial coef. The different ways in which 10 lettered pan numbers can be generated in such a way that the first five letters are capital alphabets and the next four are digits and the last is again a capital letter. Rule of sum rule of product counting integers in a range the rule of sum and the rule of product are two basic principles of counting that are. Cs 70 discrete mathematics and probability theory counting. Discrete mathematics rule of sum and rule of product duration. For example, with luck, you just stated the sum principle illustrated in figure 1.
The aim of this book is not to cover discrete mathematics in depth it should be clear. We follow a highlevel approach also adopted in most introductory textbooks in discrete mathematics as long it is well understood how we can technically formalize the arguments. Number of available phone numbers with 7 digits in the local calling area number of possible match starters football, basketball given. Lecture notes for college discrete mathematics download book. We wrap up the section on counting by doing a few practice problems and showing the intuitions behind solving each problem. Problems concerning a deck of cards are also good examples. In contrast, continuous mathematics deals with objects that vary continuously, e. Ralston, discrete algorithmic mathematics, a k peters, 3 rd edition, 2004. Practice counting possible outcomes in a variety of situations. Menu algebra 2 discrete mathematics and probability counting principle. If we can break the objects we are counting into separate, nonoverlapping disjoint cases, the total number of objects is the sum of the numbers for each individual case.
Throughout the course, students will be expected to demonstrate their understanding of discrete mathematics by being able to do each of the following. Discrete mathematics counting theory in daily lives, many a times one needs to. We now present three simple examples to illustrate this. The existence, enumeration, analysis and optimization of discrete structures. Counting theory discrete mathematics in daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. This page is dedicated to problem solving on the notions of rule of sum also known as addition principle and rule of product also known as multiplication principle. Cs 70 discrete mathematics and probability theory fall 2009 satish rao,david tse note 9 counting in the next major topic of the course, we will be looking at probability. There are nd ways to do a task if it can be done using a procedure that can be carried out in n ways, and for every way w, exactly d of the n ways correspond to way w. Discrete here is used as the opposite of continuous.
The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. We introduce the rule of sum addition rule and rule of product product rule in counting. If we are dealing with the occurrence of more than one event or activity, sometimes it is important to be able to determine how many possible outcomes exist. Pigeonhole principle states that if there are fewer pigeon holes than total. Hauskrecht counting assume we have a set of objects with certain properties counting is used to determine the number of these objects examples. Here \discrete as opposed to continuous typically also means nite, although we will consider some in nite structures as well. Discrete structures lecture notes stanford university. Dec 21, 2018 we introduce the rule of sum addition rule and rule of product product rule in counting.
Basic counting department of computer science university of. Hopefully this short introduction will shed some light on what the subject is about and what you can expect as you move. For n given weights, what is the greatest number of different. Discrete mathematics counting theory in daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. Counting onetoone functions i how manyonetoonefunctions are there from a set with 3 elements to a set with 5 elements.
Browse other questions tagged combinatorics discretemathematics or. Lets come up with an algorithm that generates a seating. We introduce some elementary concepts of combinatorics in terms of counting problems. Think of digital watches versus analog watches ones where the second hand loops around continuously without stopping.
Lecture notes principles of discrete applied mathematics. Combinatorics is the study of arrangements of objects, it is an important part of discrete mathematics. Principle of inclusionexclusion in general, we have the following rule. For solving these problems, mathematical theory of counting are used. Counting poker hands provides multiple additional examples.
Besides reading the book, students are strongly encouraged to do all the. Balakrishnan, theory and probl ems of combinatorics, schaums outline series, mcgrawhill, 1995 s. Ma8351 notes discrete mathematics regulation 2017 anna. In stage 1 we choose a person for chair 1, in stage 2 we choose a person for chair 2, etc. Here \ discrete as opposed to continuous typically also means nite, although we will consider some in nite structures as well. Discrete mathematics cmarcin sydow productand sumrule inclusionexclusion principle pigeonhole principle permutations generalised permutations andcombinations combinatorial proof binomial coe. For instance, in how many ways can a panel of judges comprising of 6 men and 4 women be chosen from among 50 men and 38 women. The basics of counting the pigeonhole principle permutations and combinations binomial coefcients and identities generalized permutations and combinations colin stirling informatics discrete mathematics chapter 6 today 2 39. Find materials for this course in the pages linked along the left. Interconnections, generalizations and specializationrelations between sev.
Ma8351 notes discrete mathematics regulation 2017 anna university free download. Discrete mathematics counting practice we wrap up the section on counting by doing a few practice problems and showing the intuitions behind solving each problem. These problems contain among them the kernels of many of the fundamental ideas of combinatorics. Discrete mathematics counting theory in discrete mathematics. Combination problems and solutions counting principle. Suppose that i love discrete math tshirts come in five different. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Menu algebra 2 discrete mathematics and probability counting principle if we are dealing with the occurrence of more than one event or activity, sometimes it is important to be able to determine how many possible outcomes exist. The discrete mathematics notes pdf dm notes pdf book starts with the topics covering logic and proof, strong induction,pigeon hole principle, isolated vertex, directed graph, alebric structers, lattices and boolean algebra, etc.
Below, we give two examples of proofs by contradiction. These problems cover everything from counting the number of ways to get dressed in the morning to counting the number of ways to build a custom pizza. Roughly, the plan is to touch on the following topics. To solve problems on this page, you should be familiar with the following notions. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. Rosen, discrete mathematics and its applications somchai prasitjutragul, discrete mathematics. The rule of sum and rule of product are used to decompose difficult counting problems into simple problems.
Think of digital watches versus analog watches ones where the. This is a course note on discrete mathematics as used in computer science. The fine art of counting lecture notes lecture notes course home. Discrete mathematics pdf notes dm lecture notes pdf. The aim of this book is not to cover discrete mathematics in. Rule of sum and rule of product problem solving brilliant. The mathematics in these applications is collectively called discrete mathematics. This is a consequence of the pigeonhole principle, but the fact. Working from basic principles and using elementary tools we develop the basic theory in its full generality. When there are m ways to do one thing, and n ways to do another, then there are m. Discrete mathematics counting theory tutorialspoint.
Share copy and redistribute the material in any medium or format adapt remix, transform, and build upon the material under the following terms. It is essential to understand the number of all possible outcomes for a series of events. Hauskrecht inclusionexclusion principle used in counts where the decomposition yields two count tasks with overlapping elements if we used the sum rule some elements would be counted twice inclusionexclusion principle. Dec 21, 2014 we wrap up the section on counting by doing a few practice problems and showing the intuitions behind solving each problem. How many bitstrings of length 8 start either with a bit 1 or end with. More formally, we begin with a basic counting principle. Discrete mathematics and probability algebra 2 mathplanet.
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