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    Discrete Mathematics With Cryptographic Applications: A Self-Teaching Guide to Unlocking the Power of Advanced Concepts and Computational Techniques
    Discrete Mathematics With Cryptographic Applications: A Self-Teaching Guide to Unlocking the Power of Advanced Concepts and Computational Techniques
    Discrete Mathematics With Cryptographic Applications: A Self-Teaching Guide to Unlocking the Power of Advanced Concepts and Computational Techniques
    Ebook559 pages6 hours

    Discrete Mathematics With Cryptographic Applications: A Self-Teaching Guide to Unlocking the Power of Advanced Concepts and Computational Techniques

    By Mercury Learning and Information and Alexander I. Kheyfits

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    This book offers a comprehensive guide to discrete mathematics and its applications to cryptography. It is designed for students and professionals in fields such as discrete mathematics and finite mathematics, with all necessary prerequisites clearly explained and illustrated. The text introduces key concepts in number theory, coding theory, and information theory, which are essential for understanding cryptography.
    Understanding discrete mathematics is crucial for anyone working in cryptography and related fields. The book begins with a survey of elementary functions and moves on to propositional algebra, set theory, and algebraic structures like groups, rings, and fields. It covers binary relations, combinatorics, and elements of number theory, which are fundamental to cryptographic methods.
    Readers will explore topics such as Boolean functions, hashing functions, cryptographic maps, combinatorial circuits, and graph theory. The book also delves into advanced areas like finite automata, game theory, and Turing machines. Through numerous examples, problems, and solutions, readers will gain a solid foundation in discrete mathematics and its cryptographic applications.

    LanguageEnglish
    PublisherPackt Publishing
    Release dateAug 2, 2024
    ISBN9781836646921
    Discrete Mathematics With Cryptographic Applications: A Self-Teaching Guide to Unlocking the Power of Advanced Concepts and Computational Techniques

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      Discrete Mathematics With Cryptographic Applications - Mercury Learning and Information

      PREFACE

      Mathematics and sciences, including political science have always involved many problems requiring large computations and sometimes, thousands of people. A century ago, the word computer meant a human being performing those computations. The top-level scientists, like Gottfried Leibniz, Lady Ada Lovelace, Alan Turing, and others were thinking how to automatize those computations. The breakthrough happened in the thirties-forties of the previous century, when Alan Turing proved the theorem about the universal program and John von Neumann developed the scheme of the universal computer machine, which all of us have used since then – see Chapter 23. The technology was ready, and the very first electronic computer ENIAC executed its initial programs in 1945. Since then, computer has meant an electronic device.

      ENIAC declassifying in 1946 led to the explosive growth of the number of electronic computers, and correspondingly, to the growth of the number of people working with computers. Those people were mostly mathematicians, but very soon mathematics departments could not provide enough cadres, and the colleges and universities had to start teaching different courses to the new computer scientists, whose first mathematical course was often called Discrete Mathematics or Finite Mathematics. Within a few years, discrete mathematics has become an independent mathematical discipline. Paul Halmos predicted [25, p. 19] that in the foreseeable future . . . discrete mathematics will be an increasingly useful tool in the attempt to understand the world, and ... analysis will therefore play a proportionally smaller role.

      Textbooks for the new discipline quickly followed, from concise lecture notes to Kenneth Rosen’s monumental treatise [43], whose 8th edition (2019) amounts to more than 1000 pages. But every new generation meets their own challenges and requires new textbooks, see, e.g., [42], whatever good are the previous ones. A quarter century back, it was important to emphasize the algorithmic nature of discrete mathematics [37]. Now the 20-30-year-old discussions – see, e.g., [36], are mostly forgotten. The Internet security issues are more important than ever before, and this textbook was written with strengthened attention to these topics. A short version of these lectures was initially taught in 1971 and has evolved during the following half-century, being greatly influenced by the times and people – by my students, friends, and colleagues on both sides of the Atlantic.

      To describe the contents of this book in more detail, let us notice that our thoughts are initially appeared as electrical-chemical potentials in the grey matter in our brains. To communicate our thoughts to other people, those potentials must be converted into electrical potentials governing oscillations of our vocal cords, and the latter create the fluctuations of the air pressure, carrying our speech. When these acoustical oscillations reach our ears, all the transformations are done in the opposite direction. Thus, the information that we create and obtain, is to be encoded and decoded in many different ways. These transformations are studied by coding theory, which is an important part of the discrete mathematics. However, to perform those conversions, we must represent the signals in a suitable form, i.e., as Boolean functions, and the book starts with an exposition of elementary logic and Boolean calculus. What is more, we often do not want other people to know the outcomes of our dealings with information, hence giving rise to the cybersecurity issues.

      Boolean functions are maps, and chapters devoted to the functions, sets, and relations follow. We also consider in more detail predicates and quantifiers, which are necessary in many applications. When it is appropriate, we include expositions of some applications of these mathematical questions to computer-related issues, such as, e.g., relational databases or hashing functions. Several classical discrete mathematics topics, namely combinatorics, graph theory, complete systems of functional elements, etc., and some applications like finite automata, necessary for the potential users of the book, are included. These developments are based upon some algebraic structures, like groups, rings, fields, and Boolean Algebras. That allowed us to consider certain cryptography issues, in particular, the Discrete Logarithm Problem.

      The book also contains chapters about number theory and game theory. No discussion of cryptography is possible these days without number-theoretical issues, like clock arithmetic and CRT. That background allows us to consider affine ciphers and some procedures of sharing secrets. The book also includes a chapter about game theory, the topic, whose inclusion in a discrete mathematics text is long overdue.

      It is not the first text, which uses cryptology to enhance the teaching of mathematics – see, e.g., an interesting note [7]. Due to the volume restrictions, specifically crypto-questions are not covered here as much as they deserve. If the discussion of crypto–issues would deviate too far from the discrete mathematics, we are to stop there. To proceed farther, we would recommend the excellent books [2, 13, 14, 40]. The table of contents shows in more detail, what is in the book.

      Many sections contain more material than can be reasonably taught at a two-hour class. That allows a lecturer some freedom of choice, and also provides the material for the student’s individual work.

      No mathematical exposition is possible without mathematical induction, which is unknown to most high school students. The author’s experience shows though, that the method of mathematical induction, being properly explained, is well accessible to college freshmen. The very first chapter of the book is devoted to mathematical induction and to a very brief discussion of elementary functions, necessary in the textbook. That sets out the prerequisite level for the whole book. It is supposed that most college freshmen, independently upon their specialization, will benefit from this textbook. However, those lower prerequisites brought an unexpected problem. Certain language, which the college sophomores usually know quite well, can be unfamiliar to some potential readers. Thus, the book explains in more detail than usual, certain parlance, e.g., necessary and sufficient conditions or the like. This is especially necessary and useful now due to proliferation of on-line courses, where the student often cannot ask immediately certain simple questions. The reader, who is fluent with that material, can skip this material with no harm.

      Of course, this is a mathematics textbook, and its reading requires concentration. To develop this culture, the reader is supposed to solve at least some of the included problems, and to analyze the suggested solutions. This is especially important in the times of online education. We remark in passing that it is not unusual, when the students find new solutions of old problems or suggest new problems. The author always welcomes any such input. The book contains more than 600 problems of various levels of difficulty. All of the exercises for the students’ individual work are placed in the end of chapters. Many other problems can be found in the cited literature, e.g., in [19].

      We thank many people for help. Parts of the book were discussed at the DIMACS Center at Rutgers University during the Reconnect conferences. The Reconnect-2019 workshop at the Champlain College in Burlington, VT, was invaluable in finalizing this project. The present author, as anyone else, has its favorite books; I cannot list all of them, but I want to mention delightful books by M. Schroeder [45] and by L. Lovasz, J. Pelikan and K. Vesztergombi [35]. We want to personally thank the following people: Midge Cozzens, Fred Roberts, David Pallai and the staff of Mercury Learning and Information, who made this project possible.

      Alexander I. Kheyfits, PhD.

      September 2021

      CHAPTER 1

      A BRIEF SURVEY OF ELEMENTARY FUNCTIONS

      1.1 MATHEMATICAL INDUCTION

      The Principle, or Axiom, or Postulate of Mathematical Induction, is one of the cornerstones of any mathematical reasoning and, in particular, of our course. It is claimed that the outstanding mathematician Leopold Kronecker (1823–1891) said that God created natural numbers, all else is humans’ business. As with any trivial truth, it can be wrong. Indeed, many thousand years ago, together with learning to talk, people started to count, and eventually, the names for small numbers had appeared, like one, two, three, etc., different for various languages. Moreover, we know, for example, from observations of Russian ethnographer and traveler Nikolas Miklukho-Maklai (1846–1888) over the indigenous people of Papua-New Guinea, that people had initially developed several specialized versions of the word one, such that the phrase one tree sounded initially differently than one boat, or than one kid. Only gradually, during millennia, those various versions of one something merged in the abstract word one, which means the natural number without any specific meaning attached.

      During the centuries, arithmetic has been developed together with the human society, and now there is the highly sophisticated mathematical discipline, the Number Theory, which studies, in particular, the properties of the natural numbers¹ {0,1,2,3,…}, of the (positive, negative, and zero) integers {…,−3,−2,−1,0,1,2,3,4,…}, of the prime numbers, etc. We accept as the known facts that the natural numbers and the integers satisfy the four standard arithmetic operations. In particular, if any three integers are connected as a = b × c, which can be written as a = b ∙ c, then the integers b and c are called factors or multipliers, and a is called the product. If we rewrite the equation as a ÷ b = c, then a is called the dividend, b the divisor, and c the quotient. It is also said that b (and c as well) divides a, or that a is divisible by b and by c, or that b (and c) divides into a.

      Definition 1 A natural number p > 1 is called prime, iff ² it has only two positive divisors, 1 and itself. For example, 3 is an improper divisor of 3 and a proper divisor of 6, but it is not a divisor of 2. A natural non-prime number p > 1 is called composite. The number 1 is neither prime nor composite, it is an improper divisor of any integer.

      Thus, the sequence of primes begins with {2,3,5,7,11,13,…}, while the initial composite numbers are {4,6,8,9,10,12,…}.

      The integers can also be classified in terms of their parity, that is, their divisibility by 2. If we divide an integer by 2, there are only two possibilities, and the remainder is either 0 or 1. The integers of the first kind, with the zero remainder, are called even numbers, the other integers are called odd numbers; thus, the integers {…,−6,−4,−2,0,2,4,6,8,…} are even, and {…,−5,−3,−1,1,3,5,7,…} are odd. Of course, the integers can be classified in terms of their divisibility over any other integer.

      Problem 1 Prove that 2 is the only even prime number.

      Proof. If p is an even prime, then p = 2p1. But if p1 > 1, then p has two factors, thus, it is not a prime number, and we arrived at the contradiction. So that, p = 2 × 1 = 2.

      This method of proof is called The Proof By Contradiction (Reductio Ad Absurdum) and is systematically used in mathematical reasoning; see Problem 52. When one wants to prove a statement by contradiction, she assumes an opposite statement and uses logical methods to deduce a statement, which contradicts the axioms or some known statements. More regarding the logic behind this and similar laws will be said in lecture 19.

      We will continue our study of the integers in Chapter 8, devoted to the Number Theory.

      Because we usually want to be correct in our statements, we must study certain properties of our reasoning, making us believe that our proofs are correct. All the more this is crucial now when we delegate some proofs to the computers and have to trust them. We discuss now only the Natural Numbers. In particular, a profound property called the Axiom or Principle of Mathematical Induction is considered at the beginning because we cannot proceed further without it.

      During millennia, people established two important, in a sense, opposite methods for proving their claims, called Deduction and Induction. Both methods are widely employed in mathematics. Employing the deduction means that we have a general statement, maybe without regard to the source of the statement, and then apply it, deduce the properties of the specific entities. Consider, for example, a plane right triangle with the acute angles 20° and 70°. Since it is said that the triangle is right, we deduce that the triangle satisfies the Pythagorean theorem, that is, its sides satisfy a² + b² = c², where a and b are the lengths of the legs and c the length of the hypotenuse. When we study mathematics, all the time, we make such deductions.

      Employing the induction method, though, we move in the opposite direction – we consider certain particular cases, analyze them, and draw a conclusion about the general case. Consider an example. Remind that the integers {2,4,6,8,10,…} are called even numbers, while the odd numbers are {1,3,5,…}.³

      Problem 2 Find the sum of n consecutive odd numbers, starting with 1.

      Solution. Mathematics is, in a sense, an experimental science. Let us perform a few numerical experiments and compute these sums. Indeed,

      1+3=4,1+3+5=9,1+3+5+7=16.

      We observe that all these sums are perfect squares, 4 = 2², , , and even the very first . The evidence is compelling, that if we add first n natural numbers, the sum is the square of their number n. To do the mathematics, let us express all that in symbols. In particular, let us notice that any odd number can be denoted as ; thus, if , then , if , then , etc. Vice versa, say implies . One can also write the odd numbers as , but here the parameter .

      There is a convenient notation for the summation formulas above, called the sigma-notation. Sigma is the name of the Greek capital letter traditionally used to denote various sums; the small sigma is . Thus, we can write

      Here, j is called the index of summation, or the summation index, or dummy index; is the lower index, is the upper index. As another example, consider . With the same success, instead of j we can choose any other convenient symbol; the only restriction is that we must avoid the collision of variables, that is, the same letter cannot stand for different variables.

      Thus, the sums of odd numbers above can be written as

      ,

      or as

      or as

      etc. As an example, we use this notation to prove that

      (1.1)

      The actual difficulty is that we do not want to establish these equations only for or , or say, for . That would be easy, just do the computations long enough. Indeed, suppose we have just finitely many statements to prove, say, we want to prove the statements , where N0 is, maybe very big, but finite natural number. We remember that the true conditional with the true premise must have the true conclusion. So that, , since and . But now from and we conclude that . By the same token, we imply next that , then , etc. Thus, after the finitely many steps, we conclude that . The actual magnitude of the integer number N0 in this reasoning is immaterial; only its finiteness is important.

      The situation is quite different if we have infinitely many statements. In this case, we physically cannot finish this domino-game and must claim certain new properties of the natural series. During the years, people separated that key property of the set of natural numbers. We must prove equation (1.1) for every natural n, that is, we have to establish it for infinitely many values of n. But our life is finite, and independently upon the speed of our computations, we cannot perform all of them in finite time. The method of proof, avoiding that infinity, is called the Principle (or the Axiom or the Postulate) of Mathematical Induction and goes in two steps as follows.

      The Axiom of Mathematical Induction. Consider a set of statements, maybe formulas , numbered by all sufficiently large integers . Usually or , but it can be any integer. The statement Sn is called the Induction Hypothesis or Inductive Assumption.

      (1) First, suppose that the statement , called the basis step of induction, or just the base is valid. In applications of the method of mathematical induction, the verification of the basic step is an independent problem. In some problems, this step may be trivial, but it can never be skipped altogether.

      (2) Second, suppose that for each integer , that is, bigger than or equal to the basis value, we can prove a conditional statement , that is, we can prove the validity of the hypothesis for each specified natural assuming the validity of Sn, and this conditional statement is valid for all natural . This part of the method is called the inductive step.

      (3) If we can independently show these two steps, then the Principle of Mathematical Induction claims that all infinitely many of the statements Sn, for all integer are valid.

      This method of proof is accepted as an axiom because nobody can actually verify infinitely many statements ; the method cannot be justified without using some other, maybe even less intuitively obvious, properties of the set of natural numbers. Mathematicians have been using this principle for centuries and never arrived at a contradiction. Therefore, we accept the method of mathematical induction without a proof, as a postulate, and believe that this principle properly expresses a certain fundamental property of the infinite set of natural numbers.

      We will apply the postulate (the axiom) of mathematical induction many times in the sequel chapters; the method will be employed in many proofs in this chapter; however, sometimes the method presents itself only implicitly, through some known results that have been already proved by the use of the mathematical induction. In the following problem, we give a detailed example of an application of the method of mathematical induction. We believe that this postulate is an intrinsic property of the set of natural numbers. When we apply this principle, we, for short, say that the proof was done by induction. First, we show how the method works in solution below, and than give the exact statement.

      Solution of Problem 2. We show now how the method works. We have already formulated the inductive assumption:

      We can choose the obvious statement as a basis of induction. That establishes the first step, the basis of induction.

      To make the inductive step, it is often useful, but not necessary! to denote the parameter as (or any other convenient character) and write down

      (1.2)

      and

      (1.3)

      To apply the inductive reasoning, we assume that the statement Sk holds good for any natural , and have to prove the validity of the statement . It is crucial in the proof of the implication that we do not use any additional properties of k: for instance, we cannot suppose that k is an odd number; the proof must work for all the natural numbers.

      Let us compare the left sides of equations (1.2) and (1.3): we observe that the former is a part of the latter. Moreover, its value is known by the inductive step; thus, we can rewrite the latter as

      Since we know that , we arrive at (1.3).

      We want to emphasize that we did not perform infinitely many computations; we replaced them by referring to the Axiom of Mathematical Induction. In what follows, we often say that we proceed by induction. Important properties of the prime numbers, in particular, their applications to cryptography, will be studied in the following chapters. Now, we consider more examples, starting with the simplest.

      Problem 3 Prove that for every natural ,

      Proof. Here and Sn stands for the equation above; thus, S1 denotes the equation , which is certainly true, S2 denotes , which is true as well; S3 is also a valid statement . Therefore, we have the basis of induction; of course, it was enough to verify only one statement S1. Now we have to validate the inductive step, that is, to prove the equation , assuming that Sn is valid for some unspecified but fixed natural . In this problem, we must prove the statement (the equation) , which reads

      assuming that Sk is valid, that is, using the equation

      as if it were correct. Its validity is unknown yet; however, in the procedure, we suppose it to be true. It is worth repeating that our reasoning must be valid for any natural number , that is, the reasoning can use only properties common to all natural numbers. For instance, we cannot assume that n is an even number.

      Observe that the left-hand side of the equation

      contains the left-hand side of Sk, and the latter in our inductive reasoning is considered to be known. This observation gives us the idea of the proof. Since we assume that the equation

      holds true, we employ Sk to transform the left-hand side of as follows,

      Thus, we have derived the statement from Sk for an arbitrary fixed natural . Since we completed both steps of the Principle of Mathematical Induction, we claim that Sn is valid for all natural n.

      Problem 4 Set

      . Prove that for any , .

      Proof. We leave a standard inductive proof to the reader, and give even simpler elementary proof without the induction to show that however powerful, the Method of Mathematical Induction is not the panacea. Indeed, since , , etc., the sum is telescoping, that is,

      Problem 5 Let the integer

      . Prove that the integer is a perfect square, and q is the smallest natural number with this property.

      This is a confirmation that the incomplete induction, that is, the reasoning, based on finitely many confirmed cases, does not work in mathematics; it can result in a wrong conclusion. Such reasoning, called incomplete induction, can lead to mistakes; it does not matter how many particular cases we have verified.

      Problem 6 Let P(x) be a quadratic polynomial . Verify that the integers are the prime numbers, but P(40) is composite.

      Proof. Now, , which is prime. For a proof by induction, it is enough to have the base of one element 0, but in this example it is interesting to check a few more cases. Thus, , , , and all these values are prime numbers. It is quite natural to assume that is a prime number for every natural k. We continue the trials, and , which are both prime numbers, thus confirming our guess. However, after that we get a surprise, since , thus, it is not prime. The latter means that the hypothesis "The number P(n) is prime for every natural n" is wrong. The number, 40 in the example, is called a counter-example to our hypothesis.

      Problem 7 Find the polynomial P(x) with integer coefficients, such that the values P(x) are prime for . What is the degree of this polynomial?

      Solution. There are infinitely many such polynomials, for example, .

      The following example shows that while performing the inductive step , one cannot omit even one step, even one value k.

      Example 1 Prove that all ladies have blue eyes.

      Proof. Of course, everyone knows at least one lady with blue eyes – we fix her, and this is the base of induction. To make the inductive step, we have to prove that any group of ladies has the blue eyes, assuming that any group of k ladies has the same blue eyes. To do that, we select any group of ladies, select any lady G1 in this group, and remove this lady from the group. By inductive assumption, the remaining k ladies all have the blue eyes. Now we return the lady G1 into the group and separate another lady . All the ladies except G2 also have blue eyes. Therefore, G1 and G2 also have the blue eyes, and all ladies have blue eyes.

      Certainly, this is a joke,⁴ and the actual problem is to find at what point this reasoning goes wrong.

      The Principle of Mathematical Induction can be stated in several equivalent forms, such as the following one.

      The Axiom of Mathematical Induction in equivalent form. Let S be some set of natural numbers. Let and if some natural number , then also the following natural number . Then S is the set of all natural numbers .

      Problem 8 Prove that in every problem above one can apply any of the equivalent forms of the axiom of induction.

      Problem 9 Prove that diagonals divide an gon, that is, the polygon with n sides (not necessarily convex) into parts.

      Proof. It is convenient in this problem to start at . Thus, a polygon is a triangle, which has no diagonal, and consists of part, that establishes the basis of induction. Now assume that for all the the statement is true, and consider any polygon P with sides. Any its diagonal d splits P into two smaller polygons with a k1 and a k2 sides, respectively, where counts d twice. Thus, . On the other hand, the total number of parts is

      .

      The following example shows that the induction can be employed to prove inequalities as well.

      Problem 10 Prove that

      Problem 11 The next is called The Strong Mathematical Induction Principle. Let S be some set of natural numbers and . Moreover, if for any natural number k, the natural numbers , then also the following natural number . Then S is the set of all natural numbers .

      The assumptions here seem to be stronger than in the standard principle above, since we assume something not only about k, but also about all smaller natural numbers . Thus the statement looks weaker. But both principles are equivalent.

      Indeed, prove that the Strong Principle of Mathematical Induction is equivalent to the Principle of Mathematical Induction in standard form.

      The natural numbers satisfy the Axiom of Mathematical Induction. Moreover, they have a more general property, the Well-Ordering Principle; it considered in more detail, for example, in [31, p. 20].

      1.2 FACTORIALS AND THE STIRLING FORMULA

      In the previous computation, we had to multiply consecutive natural numbers. This procedure occurs so often that it deserved its own name and symbol.

      Definition 2 The product of n consecutive natural numbers from 1 through n inclusive is called the n factorial and is denoted as n!.

      For example, ; if , then . If we want to preserve this property for , it is natural to define .

      The symbol n! does not look very impressive for small n but let us study it more carefully. Already , and we observe that the factorials grow very fast. James Stirling (1692–1770), the younger contemporary of Newton (1643–1727), showed the asymptotic formula to be proved in Lemma 2,

      (1.4)

      The letter e is a standard symbol for the famous real number , which will be discussed below. The symbol and name here mean that the ratio of the left- and right-hand sides of the formula tend to 1 as . This formula, without the precise value of the constant, was known to Abraham de Moivre (1667-1754) even before Stirling.

      Problem 12 Approximate 10! by formula (1.4) and compare with the exact value given above.

      Solution. By Stirling’s formula, , with the relative error less than 1%. If n is increasing, the relative error is even smaller.

      Problem 13 Compute . For what natural numbers is this expression defined?

      Solution. For , hence .

      Problem 14 How many digits are in the number 100!?

      Solution. Of course, we are not going to compute the huge number 100! precisely and count the digits. It is much easier to use a calculator and get . Hence,

      so that the decimal number 100! has 158 decimal digits.

      1.3 RECURSIVE DEFINITIONS

      The definition of the factorial function can be written as

      Such a definition is called recursive because at the second step, it returns to the same definition, but with a smaller value of the parameter. Indeed, we compute the factorial through the factorial. Recursive definitions are often used in computer science and mathematics. As another example, let us consider a recursive definition of integer powers , that can be defined for any natural n as and .

      The definitions of well-known arithmetic and geometric progressions (sequences), namely,

      where a0 is the initial term and d, called the difference, are given numbers, and

      are also recursive definitions.

      Problem 15 List the first five terms of the arithmetic progression with the first term and the difference . Prove that the terms of any arithmetic progression satisfy

      Solution. To prove the first equation, it is enough to employ the definition and iterate it, .

      Problem 16 Given , where A and B are some constants, and where ; find an explicit expression for .

      Solution. Look for the solution as , where the constants and are the roots of the quadratic equation The conditions for a0 and a1 give and .

      This solution can be easily generalized for any homogeneous difference equations with constant coefficients and with a certain non-homogeneity. The method is similar to the case of linear differential equations with constant coefficients, see, for example, [28, Sect. 4.4].

      Problem 17 Find explicitly the Fibonacci numbers, which satisfy the difference relation and the initial conditions .

      Solution. The relation and the initial conditions lead to the quadratic equation with the irrational roots and the formula for the Fibonacci numbers

      This formula, containing radicals, gives for every natural n integer values. For example, , etc.

      To prove that a recursive definition returns the quantity it is supposed to, one must use the mathematical induction. For instance, in the example above, the zeroth part of the definition, , must be used as a basis of induction. Then, the inductive step is justified by the second part of the recursive definition, . Any rigorous exposition, which includes a recursive definition or a recursive procedure, must be accompanied by an inductive proof of its validity.

      Problem 18 Find the recursive formula for the general term of a sequence, if it starts with , and every subsequent term is 3 more than twice the previous term.

      Solution. , for .

      Problem 19 Does the sequence , satisfy the recurrence relation ?

      Solution. No, since in general .

      Functions of two or more variables can also be defined recursively. A common example is the Ackermann function , defined as follows:

      Problem 20 Compute the values

      Problem 21 Give a recursive definition of the set of pairs of positive integers whose sum is odd.

      1.4 ELEMENTARY FUNCTIONS

      The next few pages contain a very brief survey of the basic elementary functions⁵ – Power, Exponential, Logarithmic, and Trigonometric Functions. If the reader is familiar with that material, she can safely skip it and go to the next lecture. However, we know from the experience that many students, especially at the community colleges, know (if any) this stuff insufficiently, that is why it is included here.

      Consider a quadratic equation . It has two real roots, . A similar equation has no real solution, but if we consider it over the larger set of complex numbers, the equation has two roots. The reason for that is that the map for real x is not a surjection, that is, given a y, we not always can return to x. This is a very common problem, and we address it now.

      First, we consider bijective maps and let be bijective. This means that for every element there exists one and only one element such that . Now we construct the map as follows. For every we set , where xy has been just defined. Since the element yx was defined uniquely, we uniquely defined the map . By our construction, the map g has the following properties.

      The domain of g is Y and the co-domain is X. For each ,

      (1.5)

      therefore,

      (1.6)

      The map g is called the inverse map of the map f (because it is unique) and is denoted . If the inverse map exists, f is called invertible. Let us repeat that the domain of f coincides with the co-domain of g and vice versa; the co-domain of f coincides with the domain of g. It is clear also that in this case, the inverse map, g is also bijective and invertible, and we can write . We restate the conclusion of this argument in the following theorem, where the uniqueness obviously follows from the construction of the inverse map.

      Theorem 1 Any bijective map has the unique inverse map, which is also bijective and satisfies the equations (1.5)–(1.6).

      Theorem 1 says, in particular, that a non-bijective function cannot have the inverse function. However, there may exist one-sided (from left or from right) invertible functions. We proceed as one often does in mathematics: namely, we use the properties, which have been proven in a particular case (in this case, equations (1.5) – (1.6) valid for bijective maps) as the definitions in general situation.

      Definition 3 Consider two arbitrary sets X and Y, and an arbitrary map ; f is called left-invertible if there exists a map , such that ; the map is called a left-inverse map of f.

      A map f is called right-invertible if there exists a map , which is called a right-inverse map for f, such that . The name (left or right) is given according to the position with respect to the given map f. Map f itself is a right-inverse for and is a left-inverse for .

      Theorem 2 Criterion of the unilateral invertibility. A map is left-invertible iff it is injective. A left-inverse map exists iff . It does not have to be unique. It is unique if either f is bijective or the domain X contains only one element x0.

      A map is right-invertible iff it is surjective. It exists iff .

      A right-inverse map may not be unique. It is unique if f is bijective or the co-domain Y is a one-element set.

      A map

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