Logic and proofs 2 course objectives at the conclusion of this course, you should be able to 1. Discrete mathematics function proof mathematics stack. Youre probably familiar with the sine and cosine functions in trigonometry, for example. Template for proofs in discrete and argumentative mathematics. For the love of physics walter lewin may 16, 2011 duration. Discrete mathematicslogicexercises wikibooks, open.
The converse of this statement is the related statement if q, then p. Before we explore and study logic, let us start by spending some time motivating this topic. What is the difference between discrete mathematics and. This approach can be a little cumbersome if one wants to learn graphs, trees, integer algorithms, etc. Symbolic logic is logic, it deals with decidability, consistency, validity, and sentences. Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. What i love about the style and format of this book is that at the start of every chapter, oscar inserts a near impossible problem and asks you to solve it. A statement and its converse do not have the same meaning.
Mathematics is the only instructional material that can be presented in an entirely undogmatic way. It contains sequence of statements, the last being the conclusion which follows from the previous statements. Discrete mathematicslogicexercises wikibooks, open books. We are going to apply the logical rules in proving mathematical theorems. Greek philosopher, aristotle, was the pioneer of logical reasoning. We talk about rules of inference and what makes a valid argument. A logical argument establishing the truth of the theorem given the truth of the axioms. Amanda nelson and steven yackel create a pizza in honor of the pigeon hole principle. It is because unless we give a specific value of a, we cannot say whether the statement is true or false. Discrete mathematics is about, well, discrete mathematics. Represent logical statements in propositional and predicate calculus, and use truth tables and formal proofs to determine their truth values. The argument is valid so the conclusion must be true if the premises are true.
First, determine what you are being asked to prove. I go off on tangents all the time looking up proofs on things that arent in line with the topic. Is discrete math the same as a philosophy logic course. The development of some material on this site was funded by nsf due0230755. Discrete math is such a broad category as to not be particularly meaningful it includes pretty much anything that can be done in a countable domain. Discrete mathematics mathematical proof techniques 1432 proof by cases i in some cases, it is very di cult to prove a theorem by applying the same argument in all cases. Mathematicians reduce problems to the manipulation of symbols using a set of rules. This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. Discrete mathematics direct proofs examples duration. Discrete math, proving a set is a subset of another. To show an object is unique one approach the one taken here is to assume that there is a second object that satisfies the given conditions.
The rest of this note covers examples of proofs by induction. An introduction to logic and proof techniques mathematics. I have made money from other survey sites but made double or triple with for the same time and effort. The content covered by most discrete math for computer science majors classes is too much to describe in one article, so well start with propositional logic. The ability to reason using the principles of logic is key to seek the truth which is our goal in mathematics. If so, i could write up a nice, detailed solution, but i wonder if the main issue is in understanding how proofs by contradiction work and are logically valid. A logical approach to discrete math texts and monographs. David griess book is great at showing the connection between logic and discrete math. How to understand and get good at proofs in discrete. Discrete mathematics by oscar levin, a fantastic book. These applets accompany the textbook, discrete mathematics.
It is also a valuable resource for professionals in various technical fields who would like. A function is, loosely defined, an operation that takes as input one or more parameter values, and produces a single, welldefined output. Note that every integer is either even or odd and no integer is both even and odd. Direct proof contrapositive proof by contradiction proof by cases 3. Discrete math was the least favorite class of my entire life. So i have tried to keep this introduction brief and i hope it will be a useful guide. Discrete mathematics propositional logic the rules of mathematical logic specify methods of reasoning mathematical statements. 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.
A proof is an argument intended to convince the reader that a general principle is true in all situations. Proof techniques, proof by contradiction, mathematical induction. Discrete mathematics function proof stack exchange. Certainly we cannot draw that conclusion from just the few above examples. See this page for a more specifically geometric concept of proofs.
For the two statements below, decide whether they are true or false. Understanding equality, or sameness, is a universal theme in all areas of mathematics. And when i say least favorite, i mean i would rather have my testicles forcefully removed from my body without the use of tools than ever sit through one of those lectures again. Discrete mathematicslogic wikibooks, open books for an. Existence proof examples show that there is a positive integer that can be written as the sum of cubes of positive integers in two different ways. Logic, proofs, and sets jwr tuesday august 29, 2000 1 logic a statement of form if p, then q means that q is true whenever p is true. At this point, we have seen a few examples of mathematical proofs. In this lesson, we will discuss what logic is and how it is used to formulate and evaluate arguments.
Euclids proofs were based on five fundamental axioms, such as the axiom that one and only straight line segment can be drawn between each pair of points. However, these types of basic probability questions just scrape the surface of discrete mathematics. Uniqueness proof, discrete math help mathematics stack exchange. Mathematical reasoning and proof with puzzles, patterns and games, by doug ensley and winston crawley, published by john wiley and sons. Nov 28, 2018 discrete mathematics with proof, second edition. This proof is an example of a proof by contradiction, one of the standard styles of mathematical proof. An indirect proof uses rules of inference on the negation of the conclusion and on some of the premises to derive the negation of a premise. A compound propositioncan be created from other propositions using logical connectives. Discrete mathematics inductive proofs saad mneimneh 1 a weird proof contemplate the following. If we are to interpret the concept based on any reasonable definition, logic is a branch of di.
We discuss modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, addition, simplification, and conjunction. Discrete mathematics for computer scienceproof wikiversity. Anybody can ask a question anybody can answer the best answers are voted up and rise to the top. This lecture now we have learnt the basics in logic. These problem may be used to supplement those in the course textbook. The author uses a range of examples to teach core concepts, while corresponding exercises allow students to apply what they learn. Its titled, discrete mathematics an open introduction by oscar levin most of us read math textbooks, but i encourage you to pick up engaging math books like this. Then if you can show that this second object is actually the first object then youve shown that all objects that satisfy the condition are identical. More direct proof examples i an integer a is called aperfect squareif there exists an integer b such that a b2. Aug 01, 2015 discrete math is such a broad category as to not be particularly meaningful it includes pretty much anything that can be done in a countable domain. For example, the statement if x 2, then x2 4 is true while its converse if x2.
Discrete mathematics, set theory, and logic discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, gene. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. This is the template for dam discrete and argumentative mathematics. Everything we know about math right now is based off of these logical proofs. Discrete mathematics with proof second edition eric gossett bethel university. Euclids axiomatic method has become the foundation of modern mathematics. 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. The amount of detail that an author supplies in a proof should depend on the audience. A logical approach to discrete math texts and monographs in. These are examples of functions that take a single number the size of an angle as an input and produce a decimal. Discrete mathematics mathematical proof techniques 3032 5. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs. Discrete mathematics combines a balance of theory and applications with mathematical rigor and an accessible writing style. Discrete mathematics propositional logic tutorialspoint.
Assuming only a basic background in calculus, discrete mathematics with proof, second edition is an excellent book for mathematics and computer science courses at the undergraduate level. Propositional logic denition apropositionis a declarative statement. Uniqueness proof, discrete math help mathematics stack. Mar 03, 2011 proof techniques, proof by contradiction, mathematical induction. Discrete mathematics achieves fascinating results using relatively simple means, such as counting. Covering combinatorics, number theory, and graph theory, this intriguing course is an ideal point of entry for you to start thinking more mathematically. It deals with continuous functions, differential and integral calculus. Learn vocabulary, terms, and more with flashcards, games, and other study tools. We will need this basic fact about the integers in some of the example proofs to follow.
594 420 1203 424 897 753 684 930 264 518 1035 1378 1209 1457 716 397 439 119 1511 584 1445 1509 1329 1460 1165 1254 44 664 1282 514 819 738 397 36 58