RHODES UNIVERSITY Grahamstown 6140, South Africa Lecture Notes CCR MAT 102 - Discrete Mathematics Claudiu C. Remsing DEPT. of MATHEMATICS (Pure and Applied) 2005 Mathematics is not about calculations but about ideas. [...] Calculations are merely a means to an end. [...] Not all ideas are mathematics; but all good mathematics must contain an idea. [...] There are [...] at least five distinct sources of mathematical ideas. They are number, shape, arrangement, movement, and chance. [...] The driving force of mathematics is problems. [...] Another important source of mathematical inspiration is examples. Ian Stewart It is not easy to say what mathematics is, but “I know it when I see it” is the most likely response of anyone to whom this question is put. The most striking thing about mathematics is that it is very different to science, and this compounds the problem of why it should be found so useful in describing and predicting how the Universe works. Whereas science is like a long text that is constantly being redrafted, updated, and edited, mathematics is entirely cumu- lative. Contemporary science is going to be proven wrong, but mathematics is not. The scientists of the past were well justified in holding na¨ıve and er- roneous views about physical phenomena in the context of the civilizations in which they lived, but there can never be any justification for establishing erro- neous mathematical results. The mechanics of Aristotel is wrong, but the geometry of Euclid is, was, and always will be correct. Right and wrong mean different things in science and mathematics. In the former, “right” means cor- respondence with reality; in mathematics it means logical consistency. John D. Barrow C.C. Remsing i Mathematics is a way of representing and explain- ing the Universe in a sym- bolic way. John D. Barrow Mathematics may be de- fined as the subject in which we never know what are we talking about, nor whether what we are say- ing is true. Bertrand Russell What is mathematics, anyway ? In a broad sense, mathematics include all the related areas which touch on quantitative, geometric, and logical themes. This includes Statistics, Computer Science, Logic, Applied Mathematics, and other fields which are frequently considered distinct from mathematics, as well as fields which study the study of mathematics (!) – History of Mathematics, Mathematics Education, and so on. We draw the line only at experimental sciences, philosophy, and computer applications. Personal perspectives vary widely, of course. Probably the only absolute definition of mathematics is : that which mathematicians do. Contrary to common perception, mathematics does not consists of “crunch- ing numbers” or “solving equations”. There are branches of mathematics con- cerned with setting up equations, or analyze their solutions, and there are ii MAT 102 - Discrete Mathematics parts of mathematics devoted to creating methods for doing computations. But there are also parts of mathematics which have nothing at all to do with numbers and equations. The current mathematics literature can be divided, roughly, into two parts : “pure” mathematics (i.e. mathematics for mathematics) and “applied” math- ematics (i.e. mathematics for something else). The first group divides roughly into just a few broad overlapping areas : Foundations : considers questions in logic or set theory – the very • language of mathematics. Algebra : is principally concerned with symmetry, patterns, discrete • sets, and the rules for manipulating arithmetic operations (one might think of this as the outgrowth of arithmetic and algebra classes in pri- mary and secondary school). Geometry : is concerned with shapes and sets, and the properties of • them which are preserved under various kinds of transformations (this is related to elementary geometry and analytic geometry). Analysis : studies functions, the real number line, and the ideas of • continuity and limit (this is the natural succesor to courses in graphing, trigonometry, and calculus). The second broad part of mathematics literature includes those areas which could be considered either independent disciplines or central parts of mathe- matics, as well as areas which clearly use mathematics but are interested in non-mathematical ideas too : Probability and Statistics : has a dual nature – mathematical and • experimental. Computer Sciences : consider algorithms and information handling. • C.C. Remsing iii Application to Sciences : significant mathematics must be devel- • opped to formulate ideas in the physical sciences (e.g. mechanics, optics, electromagnetism, relativity, astronomy, etc.), engineering (e.g. control, robotics, etc.), and other branches of science (e.g. biology, economics, social sciences). The division between mathematics and its applications is of course vague. Contents 1 Propositions and Predicates 1 1.1 Propositionsandconnectives . 2 1.2 Propositionalequivalences . 10 1.3 Predicatesandquantifiers . 14 1.4 Exercises .............................. 22 2 Sets and Numbers 27 2.1 Sets ................................. 28 2.2 Operationsonsets ......................... 34 2.3 Theintegersanddivision . 41 2.4 Exercises .............................. 54 3 Functions 60 3.1 Generalfunctions.. ... ... ... ... ... .. ... ... 61 3.2 Specificfunctions.. ... ... ... ... ... .. ... ... 74 3.3 Permutations ............................ 78 3.4 Exercises .............................. 89 4 Mathematical Induction 94 4.1 Sequencesofnumbers .. ... ... ... ... .. ... ... 95 4.2 Summations............................. 97 4.3 Mathematicalinduction . 102 4.4 Exercises .............................. 112 iv C.C. Remsing v 5 Counting 116 5.1 Basiccountingprinciples. 117 5.2 Permutationsandcombinations . 126 5.3 Binomialformula.. ... ... ... ... ... .. ... ... 136 5.4 Exercises .............................. 140 6 Recursion 144 6.1 Recursively defined sequences . 145 6.2 Modellingwithrecurrencerelations. 150 6.3 Linearrecurrencerelations. 158 6.4 Exercises .............................. 165 7 Linear Equations and Matrices 171 7.1 Systemsoflinearequations . 172 7.2 MatricesandGaussianelimination . 176 7.3 Matrixoperations . ... ... ... ... ... .. ... ... 188 7.4 Exercises .............................. 199 8 Determinants 205 8.1 Determinants ............................ 206 8.2 Propertiesofdeterminants. 210 8.3 Applications............................. 220 8.4 Exercises .............................. 225 9 Vectors, Lines, and Planes 231 9.1 Vectorsintheplane ... ... ... ... ... .. ... ... 232 9.2 Vectorsinspace. .. ... ... ... ... ... .. ... ... 244 9.3 Linesandplanes .......................... 259 9.4 Exercises .............................. 268 10 Complex Numbers 273 10.1 Numbersystems .. ... ... ... ... ... .. ... ... 274 vi MAT 102 - Discrete Mathematics 10.2 Algebraicoperations on complex numbers . 280 10.3 DeMoivre’sformula . 285 10.4 Applications... .. ... ... ... ... ... .. ... ... 291 10.5Exercises .............................. 302 A Answers and Hints to Selected Exercises 305 B Revision Problems 323 Chapter 1 Propositions and Predicates Topics : 1. Propositions and connectives 2. Propositional equivalences 3. Predicates and quantifiers Logic is the basis of all mathematical reasoning. The rules of logic specify the precise meaning of mathematical statements. Most of the definitions of formal logic have been developed so that they agree with the natural or intuitive logic used by people who have been educated to think clearly and use language carefully. The difference that exists between formal and intuitive logic are necessary to avoid ambiguity and obtain consistency. Copyright c Claudiu C. Remsing, 2005. All rights reserved. 1 2 MAT 102 - Discrete Mathematics 1.1 Propositions and connectives In any mathematical theory, new terms are defined by using those that have been previously defined. However, this process has to start somewhere. A few initial terms necessarily remain undefined. In logic, the words sentence, true, and false are initial undefined terms. Propositions 1.1.1 Definition. A proposition (or statement) is a sentence that is either TRUE or FALSE (but not both). 1.1.2 Examples. All the following sentences are propositions. (1) Pretoria is the capital of South Africa. (2) 3+3=5. (3) If x is a real number, then x2 < 0. (FALSE) (4) All kings of the United States are bald. (TRUE) (5) There is intelligent life outside our solar system. 1.1.3 Examples. The following are not propositions. (6) Is this concept important ? (7) Wow,what a day ! (8) x +1=7. (9) When the swallows return to Capistrano. (10) This sentence is false. C.C. Remsing 3 A proposition ends in a period, not a question mark or an exclamation point. Thus (6) and (7) are not propositions. (8) is not a proposition, even though it has the proper form, until the variable x is replaced by meaningful terms. Because (9) makes no sense, it cannot be TRUE or FALSE. Sentence (10) is deceiving; it looks like a proposition. If it is a proposition, then it is either TRUE or FALSE, but not both. Suppose it is TRUE. Then what it says is TRUE, and it is FALSE. But it cannot be both TRUE and FALSE. Hence (10) cannot be TRUE. Well, suppose (10) is FALSE. Then what it says is FALSE, and (10) is not FALSE, itis TRUE. Again, it cannot be both TRUE and FALSE. Therefore, (10) cannot be classified as either TRUE or FALSE. Hence it is not a proposition. We will use lower case letters, such as p,q,r,s,... to denote propositions. Any proposition symbolized by a single letter is called a primitive proposi- tion. If p is