Mathematics: A Discrete Introduction

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Click for more information on our Delivery Options. Suitable for those interested in computer science and engineering, as well as those who plan to study probability, statistics, operations research, and other areas of applied mathematics, this title teaches students the fundamental concepts in discrete mathematics and proof-writing skills.

While every attempt has been made to ensure stock availability, occasionally we do run out of stock at our stores. Prices and stock availability may vary between Webstore and our Retail Stores. Fulfilment Centre Email: sims kinokuniya. For example, a number is called prime or even provided it satisfies precise, unambiguous conditions.

These highly specific conditions are the definition for the concept. In this way, we are acting like legislators, laying down specific criteria such as eligibility for a government program.

The difference is that laws may allow for some ambiguity, whereas a mathematical definition must be absolutely clear. Definition 3. The symbol Z stands for the integers. This symbol is easy to draw, but often people do a poor job. They fall into the following trap: They first draw a Z and then try to add an extra slash. Instead, make a 7 and then an interlocking, upside-down 7 to draw Z.

Even An integer is called even provided it is divisible by two. Not entirely. The problem is that this definition contains terms that we have not yet defined, in particular integer and divisible. Each of these terms—integer, divisible, and two—can be defined in terms of simpler concepts, but this is a game we cannot entirely win.

If every term is defined in terms of simpler terms, we will be chasing definitions forever. Each part of the house is built up from previous parts. Before roofing and siding, we must build the frame. Before the frame goes up, there must be a foundation. As house builders, we think of pouring the foundation as the first step, but this is not really the first step. We also have to own the land and run electricity and water to the property.

For there to be water, there must be wells and pipes laid in the ground.

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We have descended to a level in the process that really has little to do with building a house. Yes, utilities are vital to home construction, but it is not our job, as home builders, to worry about what sorts of transformers are used at the electric substation! Let us return to mathematics and Definition 3. It is possible for us to define the terms integer, two, and divisible in terms of more basic concepts.

It takes a great deal of work to define integers, multiplication, and so forth in terms of simpler concepts. What are we to do? Ideally, we should begin from the most basic mathematical object of all—the set—and work our way up to the integers. Although this is a worthwhile activity, in this book we build our mathematical house assuming the foundation has already been laid. Where shall we begin? What may we assume? In this book, we take the integers as our starting point.

The integers are the positive whole numbers, the negative whole numbers, and zero. That is, the set of integers, denoted by the letter Z, is Z D f: : : ; 3; 2; 1; 0; 1; 2; 3; : : :g : We also assume that we know how to add, subtract, and multiply, and we need not prove basic number facts such as 3 2 D 6. See Appendix D for more details on what you may assume. Thus, in Definition 3. However, we still need to define what we mean by divisible.

To underscore the fact that we have not made this clear yet, consider the question: Is 3 divisible by 2? We want to say that the answer to this question is no, but perhaps the answer is yes since 3 2 is 1 So it is possible to divide 3 by 2 if we allow Copyright Cengage Learning. Note further that in the previous paragraph we were granted basic properties of addition, subtraction, and multiplication, but not—and conspicuous by its absence—division.

Thus we need a careful definition of divisible.

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We say that a is divisible by b provided there is an integer c such that bc D a. We also say b divides a, or b is a factor of a, or b is a divisor of a. The notation for this is bja. This definition introduces various terms divisible, factor, divisor, and divides as well as the notation bja. Example 3. To answer this question, we examine the definition. It says that a D 12 is divisible by b D 4 if we can find an integer c so that 4c D Of course, there is such an integer, namely, c D 3. In this situation, we also say that 4 divides 12 or, equivalently, that 4 is a factor of We also say 4 is a divisor of The notation to express this fact is 4j On the other hand, 12 is not divisible by 5 because there is no integer x for which 5x D 12; thus 5j12 is false.

Now Definition 3. The number 12 is even because 2j12, and we know 2j12 because 2 6 D On the other hand, 13 is not even, because 13 is not divisible by 2; there is no integer x for which 2x D Note that we did not say that 13 is odd because we have yet to define the term odd. All we can say at this point is that 13 is not even. That being the case, let us define the term odd. Thus 13 is odd because we can choose x D 6 in the definition to give 13 D 2 6 C 1. Note that the definition gives a clear, unambiguous criterion for whether or not an integer is odd.

Please note carefully what the definition of odd does not say: It does not say that an integer is odd provided it is not even. This, of course, is true, and we prove it in a subsequent chapter. Here is a definition for another familiar concept. For example, 11 is prime because it satisfies both conditions in the definition: First, 11 is greater than 1, and second, the only positive divisors of 11 are 1 and However, 12 is not prime because it has a positive divisor other than 1 and itself; for example, 3j12, 3 6D 1, and 3 6D Is 1 a prime?

To see why, take p D 1 and see if p satisfies the definition of primality. The second condition is satisfied: the only divisors of 1 are 1 and itself. Therefore, 1 is not a prime. We have answered the question: Is 1 a prime? I will attempt to answer this question in a moment, but there is an important philosophical point that needs to be underscored. The decision to exclude the number 1 in the definition was deliberate and conscious.

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The main problem with your using a different definition for prime is that the concept of a Copyright Cengage Learning. Licensed to: CengageBrain User 6 Chapter 1 Fundamentals prime number is well established in the mathematical community. If it were useful to you to allow 1 as a prime in your work, you ought to choose a different term for your concept, such as relaxed prime or alternative prime. Now, let us address the question: Why did we write Definition 3. Later, we prove the fact that every positive integer can be factored in a unique fashion into prime numbers. For example, 12 can be factored as 12 D 2 2 3.

There is no other way to factor 12 down to primes other than rearranging the order of the factors. The prime factors of 12 are precisely 2, 2, and 3. Since we have defined prime numbers, it is appropriate to define composite numbers. Similarly, the number is composite. Prime numbers are not composite. If p is prime, then, by definition, there can be no divisor of p between 1 and p read Definition 3. Furthermore, the number 1 is not composite. Poor number 1! It is neither prime nor composite! There is, however, a special term that is applied to the number 1—the number 1 is called a unit.

Recap In this section, we introduced the concept of a mathematical definition. Please determine which of the following are true and which are false; use Definition 3. Here is a possible alternative to Definition 3. Explain why this alternative definition is different from Definition 3. So, to answer this question, you should find integers a and b such that a is divisible by b according to one definition, but a is not divisible by b according to the other definition.

None of the following numbers is prime. Explain why they fail to satisfy Definition 3. Which of these numbers is composite? The natural numbers are the nonnegative integers; that is, N D f0; 1; 2; 3; : : :g: The symbol Q stands for the rational numbers. Note: Many authors define the natural numbers to be just the positive integers; for them, zero is not a natural number. To me, this seems unnatural. The concepts positive integers and nonnegative integers are unambiguous and universally recognized among mathematicians. The set of all rational numbers is denoted Q. Explain why every integer is a rational number, but not all rational numbers are integers.

Define what it means for an integer to be a perfect square. For example, the integers 0, 1, 4, 9, and 16 are perfect squares. Your definition should begin An integer x is called a perfect square provided. Define what it means for one number to be the square root of another number. Define the perimeter of a polygon. Suppose the concept of distance between points in the plane is already defined. Write a careful definition for one point to be between two other points.

Your definition should begin Suppose A; B; C are points in the plane.

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We say that C is between A and B provided. Note: Since you are crafting this definition, you have a bit of flexibility. Consider the possibility that the point C might be the same as the point A or B, or even that A and B might be the same point. Personally, if A and C were the same point, I would say that C is between A and B regardless of where B may lie , but you may choose to design your definition to exclude this possibility. Whichever way you decide is fine, but be sure your definition does what you intend.

Note further: You do not need the concept of collinearity to define between. Once you have defined between, please use the notion of between to define what it means for three points to be collinear. We say that they are collinear provided. Note even further: Now if, say, A and B are the same point, you certainly want your definition to imply that A, B, and C are collinear. Define the midpoint of a line segment.

Some English words are difficult to define with mathematical precision for example, love , but some can be tightly defined. Try writing definitions for these: a. You may assume more basic concepts such as coin or pronunciation are already defined. Discrete mathematicians especially enjoy counting problems: problems that ask how many. Here we consider the question: How many positive divisors does a number have? For example, 6 has four positive divisors: 1, 2, 3, and 6. How many positive divisors does each of the following have? Why do 30 and 42 have the same number of positive divisors?

An integer n is called perfect provided it equals the sum of all its divisors that are both positive and less than n. For example, 28 is perfect because the positive divisors of 28 are 1, 2, 4, 7, 14, and There is a perfect number smaller than Find it.


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Write a computer program to find the next perfect number after At a Little League game there are three umpires. One is an engineer, one is a physicist, and one is a mathematician. There is a close play at home plate, but all three umpires agree the runner is out. The notion of proof is the subject of the next section—indeed, it is a central theme of this book. Suffice it to say for now that a proof is an essay that incontrovertibly shows that a statement is true. In this section we focus on the notion of a theorem.

Reiterating, a theorem is a declarative statement about mathematics for which there is a proof. What is a declarative statement? In everyday English we utter many types of sentences. Some sentences are questions: Where is the newspaper? Other sentences are commands: Come to a complete stop. Practitioners of every discipline make declarative statements about their subject matter.

Such statements fall into three categories: Please be sure to check your own work for nonsensical sentences. This type of mistake is all too common. Think about every word and symbol you write. Ask yourself, what does this term mean? Do the expressions on the left and right sides of your equations represent objects of the same type? Statements we know to be true because we can prove them—we call these theorems. Statements whose truth we cannot ascertain—we call these conjectures.

Statements that are false—we call these mistakes! There is one more category of mathematical statements. We therefore call such statements nonsense! The Nature of Truth To say that a statement is true asserts that the statement is correct and can be trusted. However, the nature of truth is much stricter in mathematics than in any other discipline. Does this mean that every day in every July is hot and humid? No, of course not. It is not reasonable to expect such a rigid interpretation of a general statement about the weather. First, the value 9. Second, the term near is vague. Even at an altitude of meters, gravity is slightly less than at the surface.

Worse yet, gravity at the surface is not constant; the gravitational pull at the top of Mount Everest is a bit smaller than the pull at sea level! As climatologists or physicists, we learn the limitations of our notion of truth. Most statements are limited in scope, and we learn that their truth is not meant to be considered absolute and universal.

However, in mathematics the word true is meant to be considered absolute, unconditional, and without exception. Let us consider an example. Perhaps the most celebrated theorem in geometry is the following classical result of Pythagoras. Theorem 4. We know this because we can prove this theorem more on proofs later. Is the Pythagorean Theorem really absolutely true? We might wonder: If we draw a right triangle on a piece of paper and measure the lengths of the sides down to a billionth of an inch, would we have exactly a2 C b 2 D c 2?

Probably not, because a drawing of a right triangle is not a right triangle! A drawing is a helpful visual aid for understanding a mathematical concept, but a drawing is just ink on paper. The number 2 is prime but not odd. Therefore, the statement is false.

We might like to say it is nearly true since all prime numbers except 2 are odd. A statement that is not absolutely true in this strict way is called false. An engineer, a physicist, and a mathematician are taking a train ride through Scotland. They happen to notice some black sheep on a hillside. All we can say is that in this part of Scotland there are some black sheep. If-Then Consider the mathematical and the ordinary usage of the word prime. Mathematicians use the English language in a slightly different way than ordinary speakers.

We give certain words special meanings that are different from that of standard usage. Mathematicians take standard English words and use them as technical terms. We give words such as set, group, and graph new meanings. We also invent our own words, such as bijection and poset. All these words are defined later in this book. Not only do mathematicians expropriate nouns and adjectives and give them new meanings, we also subtly change the meaning of common words, such as or, for our own purposes.

While we may be guilty of fracturing standard usage, we are highly consistent in how we do it. I call such altered usage of standard English mathspeak, and the most important example of mathspeak is the if-then construction. Only one consequence is promised. But he also understands that if he does finish his lima beans, then he will get dessert.

The sentence does not rule out the possibility of x C y being even despite x or y not being even. Indeed, if x and y are both odd, we know that x C y is also even. Let us summarize this in a chart. Suppose I am elected and I lower taxes. Certainly you would not call me a liar—I kept my promise. Suppose I am elected and I do not lower taxes.

Now you have every right to call me a liar—I have broken my promise. Suppose I am not elected, but somehow say, through active lobbying I manage to get taxes lowered. You certainly would not call me a liar—I have not broken my promise. Finally, suppose I am not elected and taxes are not lowered. Again, you would not accuse me of lying—I promised to lower taxes only if I were elected.

It would be tiresome to use the same phrases over and over in mathematical writing. The way to understand this wording is as follows: In order for A to be true, it is necessarily the case that B is also true. This statement is verbose. The key phrase is if and only if. What does an if-and-only-if statement mean? Thus the two conditions A and B must be both true or both false. An integer x is even if and only if x C 1 is odd. Just as there are many ways to express an if-then statement, so too are there several ways to express an if-and-only-if statement. The reason for the word equivalent is that condition A holds under exactly the same circumstances under which condition B holds.

The symbol is an amalgamation of the symbols H and H. Mathematicians use the words and, or, and not in very precise ways. The mathematical usage of and and not is essentially the same as that of standard English. The usage of or is more idiosyncratic. The use of and can be summarized in the following chart.

Again, we can summarize the use of not in a chart. A True False Mathematical use of or. The use of or, however, does not. In standard English, or often suggests a choice of one option or the other, but not both. In contradistinction, the mathematical or allows the possibility of both.

For example, consider the following: Suppose x and y are integers with the property that xjy and yjx. The conclusion of this result says that we may have any one of the following: x D y but not x D y e.

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Here is a chart for or statements. A theorem is a specific statement that can be proved. A theory is a broader assembly of ideas on a particular issue. Some theorems are more important or more interesting than others. There are alternative nouns that mathematicians use in place of theorem.

Each has a slightly different connotation. The word theorem conveys importance and generality. The Pythagorean Theorem certainly deserves to be called a theorem. Here we list words that are alternatives to theorem and offer a guide to their usage. Result A modest, generic word for a theorem. Proposition A minor theorem. A proposition is more important or more general than a fact but not as prestigious as a theorem.

Lemma A theorem whose main purpose is to help prove another, more important theorem. Some theorems have complicated proofs. Often one can break down the job of proving a such theorems into smaller parts. The lemmas are the parts, or tools, used to build the more elaborate proof. Corollary A result with a short proof whose main step is the use of another, previously proved theorem.

Claim Similar to lemma. A claim is a theorem whose statement usually appears inside the proof of a theorem. The purpose of a claim is to help organize key steps in a proof. Also, the statement of a claim may involve terms that make sense only in the context of the enclosing proof. Vacuous Truth What are we to think of an if-then statement in which the hypothesis is impossible?

Consider the following. Statement 4. Is this statement true or false? The statement is not nonsense. The terms perfect square see Exercise 3. We might be tempted to say that the statement is false because square numbers and prime numbers cannot be negative. In the case of Statement 4.

So we can never find an integer that renders condition A true and condition B false. Therefore, Statement 4. Recap This section introduced the notion of a theorem: a declarative statement about mathematics that has a proof. We discussed the absolute nature of the word true in mathematics. We examined the if-then and if-and-only-if forms of theorems, as well as alternative language to express such results.

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We clarified the way in which mathematicians use the words and, or, and not. We presented a number of synonyms for theorem and explained their connotations. Finally, we discussed vacuous if-then statements and noted that mathematicians regard such statements as true. Each of the following statements can be recast in the if-then form.

The product of an odd integer and an even integer is even. The square of an odd integer is odd. The square of a prime number is not prime. The product of two negative integers is negative. This, of course, is false. The diagonals of a rhombus are perpendicular. Congruent triangles have the same area. The sum of three consecutive integers is divisible by three. Below you will find pairs of statements A and B. For each pair, please indicate which of the following three sentences are true and which are false: If A, then B. If B, then A. A if and only if B.

A: Joe is a grandfather. B: Joe is male. A: Ellen resides in Los Angeles. B: Ellen resides in California. A: This year is divisible by 4. B: This year is a leap year. For the remaining items, x and y refer to real numbers. It is a common mistake to confuse the following two statements: a. If A, then B.

Find two conditions A and B such that statement a is true but statement b is false. Consider the two statements a. Under what circumstances are these statements true? When are they false? Explain why these statements are, in essence, identical. If not B , then not A. A iff B. Under what circumstances are they false? Consider an equilateral triangle whose side lengths are a D b D c D 1. Notice that in this case a2 Cb 2 6D c 2. Explain why this is not a violation of the Pythagorean Theorem. Explain how to draw a triangle on the surface of a sphere that has three right angles. Do the legs and hypotenuse of such a right triangle satisfy the condition a2 C b 2 D c 2?

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