It has many practical applications in computer science like design of computing. Discrete mathematics i computer science tripos, part 1a paper 1 natural sciences tripos, part 1a, computer science option politics, psychology and sociology, part 1, introduction to computer science option 2012 lecturer. Richard mayr university of edinburgh, uk discrete mathematics. Greek philosopher, aristotle, was the pioneer of logical reasoning. Since spring 20, the book has been used as the primary textbook or a supplemental resource at multiple universities around the world see the partial adoptions list. Pearson discrete mathematics, 8e richard johnsonbaugh. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematical proofs.
Discrete mathematics for computer scienceproof wikiversity. The argument is valid if the premises imply the conclusion. Proofs typically begin with the word proof, and end with some sort of indication like q. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin.
Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs. This page intentionally left blank university of belgrade. Wuct121 logic tutorial exercises solutions 2 section 1. Discrete mathematics and its applications 2020spring. Form given hypothesis 1 hypothesis 2 hypothesis n then. However, the rigorous treatment of sets happened only in the 19th century due to the german math ematician georg cantor. The twosemester discrete math sequence covers the mathematical topics most directly related to computer science. But avoid asking for help, clarification, or responding to other answers. The concept of a function is extremely important in discrete mathematics. Logic and proof the rules of logic specify the precise meanings of mathematical statements.
The ability to reason using the principles of logic is key to seek the truth which is our goal in mathematics. Ma8351 syllabus discrete mathematics regulation 2017. Write these propositions using p, q, and r and logical connectives including negations. Discrete mathematics is the part of mathematics devoted to the study of discrete as opposed to continuous objects. Discrete mathematics proof by contraposition duration. We hope that these notes will prepare a student to better understand basic. Rules of inference and logic proofs millersville university. In this case we prove the property only for integers b instead of for all of n. Wuct121 discrete mathematics logic tutorial exercises.
In such a case, the basis step begins at a starting point b where b is an integer. Finish up propositional logic and start on firstorder logic. Exercise sets features a large number of applications. Discrete mathematics logic tutorial exercises solutions 1. This lecture now we have learnt the basics in logic.
Logic logic is commonly known as the science of reasoning. Spiral workbook for discrete mathematics open textbook. In particular, this class is meant to introduce logic, proofs, sets, relations, functions, counting, and probability, with an emphasis on applications in computer science. We are going to apply the logical rules in proving mathematical theorems. The vocabulary includes logical words such as or, if, etc. Notice the pattern of ts and fs in the first two columns of each of the truth tables above. This is a text that covers the standard topics in a sophomorelevel course in discrete mathematics. More than 100 new exercises have been added to the first three chapters. By denition, computers operate on discrete data binary strings. Discrete mathematics lecture notes, yale university, spring 1999 l.
Proofs by mathematical induction do not always start at the integer 0. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion is true under the assumption that the premisses are true. It deals with continuous functions, differential and integral calculus. Prior to his 20year service at depaul university, he was a member and sometime chair of the mathematics departments at morehouse college and chicago state univer. Mathematical proofs can themselves be represented formally as discrete structures. In either view, we noticed that mathematical statements have a particular logical form, and analyzing that form can help make sense of the statement. Combinatorics, graph theory, the idea of function, recurrence relations, permutations, and set theory are all part of discrete math.
An introduction to logic and proof techniques mathematics. This course is a prerequisite for all theory courses as well as discrete mathematics ii, algorithms i, security part ib and part ii, arti. Welcome to discrete mathematics 1, a course introducing set theory, logic, functions, relations, counting, and proofs. For example, if i told you that a particular realvalued function was continuous on the interval \0,1\text,\ and \f0 1\ and \f1 5\text,\ can we conclude that there is some point between \0,1\ where the.
These are notes on discrete mathematics for computer scientists. Thanks for contributing an answer to mathematics stack exchange. This course will roughly cover the following topics and speci c applications in computer science. Thus a proof is a sequence of steps linked together by modus ponendo ponens. These problem may be used to supplement those in the course textbook. Sequences and series are among the most important applications of. Introduction to logic using propositional calculus and proof 1. Next come chapters on logic, counting, and probability. Is there a discrete mathematics for dummies kind of book. In mathematics, a statement is not accepted as valid or correct unless it is accompanied by a proof. Propositional logic richard mayr university of edinburgh, uk richard mayr university of edinburgh, uk discrete mathematics. Arguments in propositional logic a argument in propositional logic is a sequence of propositions. Discrete mathematics propositional logic the rules of mathematical logic specify methods of reasoning mathematical statements.
Propositional logic propositional equivalences predicates and quantifiers nested quantifiers rules of inference introduction to proofs proof. Vesztergombi parts of these lecture notes are based on l. This is a book about discrete mathematics which also discusses mathematical rea. The natural numbers are designed for counting discrete quantities, but they suffer. Text from oscar levins discrete mathematics text cc by. Direct proof and counterexample 1 in this chapter, we introduce the notion of proof in mathematics. Most of the problems are from discrete mathematics with applications by h. Supporting our customers during coronavirus covid19. Richard johnsonbaugh is professor emeritus of computer science, telecommunications and information systems, depaul university, chicago. Grassmann and others published logic and discrete mathematics. Introduction to logic and set theory 202014 bgu math.
So, in some sense, the topics in this class are more relavent to cse major than calculus. Some of the reasons to study logic are the following. Direct proof contrapositive proof by contradiction proof by cases 3. Discrete mathematics consists of many individual topics that, imprecisely, contrast with continuous mathematics, e.
This zybook demonstrates how to translate english descriptions of everyday scenarios into precise mathematical statements that can then be used for formal analysis. Apr, 2016 discrete math might be the first proof based course you have ever taken. A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. Given a few mathematical statements or facts, we would like to be able to draw some conclusions. In case 2, x2 is also positive thus, we obtain the same conclusion in all cases, so that the original statement is true. Number theory athe math behind the rsa crypto system. Spence illinois state university charles vanden eynden. Discrete math concerns counting, probability, sophisticated forms of addition, and limit processes over discrete sets.
Cantor developed the concept of the set during his study of the trigonometric series, which is now known as the limit point or the derived set operator. A proof is an argument from hypotheses assumptions to a conclusion. Johnsonbaugh, discrete mathematics, 8th edition pearson. Text from oscar levins discrete mathematics text cc bysa. This section contains a unit on proofs, proof methods, the well ordering principle, logic and propositions, quantifiers and predicate logic, sets, binary relations, induction, state machines invariants, recursive definition, and infinite sets. Now we will go over some of the basic methods of proving an implication. Slides of the diagrams and tables in the book in both pdf and latex can be down. A proof is a valid argument that establishes the truth of a theorem as the conclusion statements in a proof can include the axioms. Mathematical logic and proofs mathematics libretexts. Discrete mathematics syllabus ma8351 pdf free download. This insistence on proof is one of the things that sets mathematics apart from other. Discrete mathematics discrete mathematics study of mathematical structures and objects that are fundamentally discrete rather than continuous. An argument in propositional logic is a sequence of propositions. With nearly 4,500 exercises, discrete mathematics provides ample opportunities for students to practice, apply, and demonstrate conceptual understanding.
An argument form is an argument that is valid no matter what propositions are substituted into its propositional variables. Discrete mathematics, set theory, and logic discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, gene. These words have very precise meanings in mathematics which can di. Discrete mathematics for computer science some notes arxiv. The zybooks approach less text doesnt mean less learning. The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics.
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. Discrete mathematics richard johnsonbaugh download. In so far as possible we try to work in the formal methods indirectly, as we take the reader through some interesting mathematics. Nevertheless, we can easily prove the implication is indeed true. Logic is the study of the principles of reasoning, especially of the structure of propositions as distinguished. In studying mathematical logic we shall not be concerned with the truth value of.
Discrete mathematics with proof, second edition continues to facilitate an uptodate understanding of this important topic, exposing readers to a wide range of modern and technological applications. Discrete mathematics pdf notes dm lecture notes pdf. An open introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math majors, especially those who will go on to teach. Nov 28, 2018 discrete mathematics with proof, second edition. Common mistakes in proofs show that if x is real number, then x2 is positive. This webpage was created as a resource for students of discrete mathematics, either those using one of my textbooks, discrete mathematics with applications, 4 th edition, or discrete mathematics. The emphasis here will be on logic as a working tool. Problems on discrete mathematics1 ltex at january 11, 2007. Discrete math for computer science students ken bogart dept. A visually animated interactive introduction to discrete mathematics.
Mathematics is the only instructional material that can be presented in an entirely undogmatic way. Strong emphasis on reading and writing proofs illustrates most proofs of theorems with annotated figures to provide additional explanation and insight into the proofs. Methods of proving common mistakes in proofs strategies. At the hardware level the design of logic circuits to implement in. Csc 201 discrete mathematics for computer scientists. An accessible introduction to the topics of discrete math, this bestselling text also works to expand students mathematical maturity. Below, you will find the videos of each topic presented. Before we explore and study logic, let us start by spending some time motivating this topic.
Math 215 discrete mathematics worksheets logic and proof. An introduction to mathematical reasoning, or those using any other textbook. Unit i logic and proofs ma8351 syllabus discrete mathematics. A mathematical rule of inference is a method for deriving a new statement that may depend on inferential rules of a mathematical system as well as on logic. We will develop some of the symbolic techniques required for computer logic. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Aims this course will develop the intuition for discrete mathematics reasoning involving numbers and sets. Quantifiers, start on inference and proofs pdf, pptx note. Perform operations on discrete structures such as sets, functions, relations or sequences. Find materials for this course in the pages linked along the left. Discrete mathematics propositional logic tutorialspoint. They are not guaranteed to be comprehensive of the material covered in the course.
But a sharp contrast between discrete and continuous mathematics is mainly a convenience. Logical reasoning provides the theoretical base for many areas of mathematics and consequently computer science. Math 215 discrete mathematics worksheets logic and proof let p, q, and r be the propositions p. The argument may use other previously established statements, such as theorems. It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its. Computer science programs often feature this course in the first or second year of university and many students find it incredibly difficult. Discrete mathematics, second edition in progress upenn cis. A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.
Calculus deals with continuous objects and is not part of discrete mathematics. He was solely responsible in ensuring that sets had a home in mathematics. Here you can download the free lecture notes of discrete mathematics pdf notes dm notes pdf materials with multiple file links to download. The rules of mathematical logic specify methods of reasoning mathematical statements. All but the nal proposition in the argument are called premises or hypotheses and the nal proposition is called the conclusion. Discrete mathematicslogic wikibooks, open books for an. This is a course on discrete mathematics as used in computer science. By grammar, i mean that there are certain commonsense principles of logic, or proof techniques, which you can. This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. Examples of objectswith discrete values are integers, graphs, or statements in logic. In the first column the truth values of p, there are 2 ts followed by 2 fs. We have considered logic both as its own subdiscipline of mathematics, and as a means to help us better understand and write proofs. Johnsonbaugh, discrete mathematics, global edition pearson.
Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. For example, if we have a finite set of objects, the function can be defined as a list of ordered pairs having these objects, and can be presented as a complete list of those pairs. Each step of the argument follows the laws of logic. Mathematical logic for computer science is a mathematics textbook, just as a. Sets and logic, proofs, and functions, sequences, and relations.
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