This book has been designed to meet the needs of almost any type of introductory discrete mathematics course. Applications and Modeling: Discrete mathematics has applications in almost every possible field of study.
Features of the Book
Exercises that develop the results used in the text are clearly identified by the right-hand symbol. All are designed to expose students to ideas not covered in depth in the text.
How to Use This Book
SUPPLEMENTARY EXERCISE SETS Each chapter is followed by a rich and varied set of supplementary exercises. These exercises are generally more difficult than those in the exercise sets that follow the sections.
Ancillaries
A course with a strong computer science emphasis may be offered by covering some or all of the optional computer science sections. Instructors can find sample syllabi for a wide range of discrete mathematics courses and teaching suggestions for using each section of the text can be found in the Instructor's Resource Guide available on the website for this book.
Acknowledgments
I thank Paul Mailhot of PreTeX, Inc., the compiler, for the tremendous amount of work he devoted to the production of this edition, and for his intimate knowledge of LaTeX. I also express my appreciation to the Science, Engineering and Mathematics (SEM) Division of McGraw-Hill Higher Education for their valuable support of this new edition and the accompanying media content.
THE INFORMATION CENTER
The extensive companion website that accompanies this text has been significantly enhanced for the seventh edition. This website is accessible at www.mhhe.com/rosen. The home page shows the Information Center and contains login links for the student site and the site's instructor site.
STUDENT SITE
This guide includes a detailed list of common misconceptions students of discrete mathematics often have and the types of mistakes they tend to make. The mistakes that students have made while studying discrete mathematics using this text have been analyzed to design this resource.
INSTRUCTOR SITE
Propositional Logic
In this chapter, we will explain what constitutes a correct mathematical argument and introduce tools for constructing these arguments. After introducing many different proof methods, we will introduce several strategies for constructing proofs.
Introduction
EXAMPLE 5 Find the conjunction of propositionsspandq, where the proposition "Rebecca's PC has more than 16 GB of free hard disk space" and q is the proposition "The processor in Rebecca's PC runs faster than 1 GHz.". Thus, a disjunction is true when at least one of the two propositions in it is true.
Conditional Statements
Our definition of a conditional statement specifies its truth values; it is not based on English usage. Recall that the contrapositive, but neither the converse nor the converse, of a conditional statement is equivalent to it.
Truth Tables of Compound Propositions
EXAMPLE 10 Let's use the statement "You can take the flight" and the statement "You buy a ticket." This statement is true if pandq is both true or both false, that is, if you buy a ticket and can take the flight, or if you don't buy a ticket and can't take the flight.
Precedence of Logical Operators
Logic and Bit Operations
Once this is done, operations on the bit strings can be used to manipulate this information. Here, and in this book, bit strings are broken down into blocks of four bits to make them easier to read.
Exercises
Applications of Propositional Logic
Additionally, propositional logic and its rules can be used to design computer circuits, build computer programs, verify the correctness of programs, and build expert systems. We will discuss some of these applications of propositional logic in this section and later chapters.
Translating English Sentences
You can only access the Internet from campus if you are a computer science major or not a freshman." You cannot ride the roller coaster if you are under 4 feet tall unless you are over 16 years old.
System Specifications
Letp means "Diagnostic message buffered" and q means "Diagnostic message resent". The specifications can then be written as p∨q,¬p and p→q. EXAMPLE 5 Does the system specification in Example 4 remain consistent if the "Diagnostic message is not resent" specification is added.
Boolean Searches
Solution: To determine whether these specifications are consistent, we first express them using logical expressions. Solution: By reasoning in Example 4, the three specifications from this example are true only if is false and q is true.
Logic Puzzles
No” the first time the question is asked, because everyone sees mud on the other child's forehead. After the son answers “No” to the first question, the daughter can determine that this must be true.
Logic Circuits
Propositional Equivalences
An important type of step used in a mathematical argument is to replace one statement with another statement with the same truth value. Because of this, methods that produce statements with the same truth value as a given compound proposition are widely used in the construction of mathematical arguments.
Logical Equivalences
We begin our discussion by classifying compound propositions according to their possible truth values. Since the truth values of the composite propositions ¬(p∨q)and¬p∧ ¬q coincide for all possible combinations of the truth values of pandq, it follows that ¬(p∨q)↔(¬p∧ ¬q) is a tautology and that these compound propositions logically equivalent.
Using De Morgan’s Laws
Heather goes to the concert or Steve goes to the concert” can be represented by r∨s. Hence, we can express the negation of our original statement as “Heather will not go to the concert and Steve.
Constructing New Logical Equivalences
EXAMPLE 5 Use De Morgan's Laws to express the negations "Miguel has a cell phone and he has a laptop" and "Heather will go to the concert or Steve will go to the concert.". The letter will be "Heather will go to the concert" and it will be "Steve will go to the concert." Then.
Applications of Satisfiability
After listing these statements, we will explain how to construct the statement that each row contains every integer from 1 to 9. We let the construction of the other statements be that each column contains each number and each of the nine 3×3 blocks contains each number. number for the exercises.
Solving Satisfiability Problems
Predicates and Quantifiers
We will see how predicate logic can be used to express the meaning of a wide range of statements in mathematics and computer science in ways that allow us to reason and explore relationships between objects. We then introduce the notion of quantifiers, which allow us to reason with statements asserting that a certain property holds for all objects of a certain type, and with statements asserting the existence of an object with a certain property.
Predicates
Similarly, we can let R(x, y, z) denote the statement "x+y=z." When values are assigned to the variables x, y, and z, this statement has a truth value. Because we want to verify that the program swaps the values afxandy for all input values, for the postcondition we can use Q(x, y), where Q(x, y) is the statement "x = band y=a.".
Quantifiers
EXAMPLE 13 What is the truth value of ∀x(x2≥x) if the domain consists of all real numbers. What is the truth value of this statement if the domain consists of all integers.
Quantifiers with Restricted Domains
However, there is no limit to the number of different quantities we can define, such as "there are exactly two", "there are no more than three", "there are at least 100", etc. Other phrases for quantifying uniqueness include "there is exactly one" and "there is one and only one.").
Precedence of Quantifiers
For example, ∃!x(x−1=0), where the domain is the set of real numbers, says that there exists a unique real numberx such that x−1=0. Note that we can use quantifiers and propositional logic to express uniqueness (see Exercise 52 in Section 1.5), so the quantity uniqueness can be avoided.
Binding Variables
The scope of the first quantifier, ∃x, is the expression P (x)∧Q(x), because ∃xi applies only to P (x)∧Q(x) and not to the rest of the statement. Note that our statement could be written using two different variables, x and y, as∃x(P (x)∧Q(x))∨ ∀yR(y), because the ranges of the two quantifiers do not overlap.
Logical Equivalences Involving Quantifiers
Consequently, a variable is free if it is outside the scope of all quantifiers in the formula specifying this variable. EXAMPLE 18 In the statement∃x(x+y=1), the variable x is bound by the existential quantifier∃x, but the variable is free because it is not bound by a quantifier and no value is assigned to this variable .
Negating Quantified Expressions
Next, note that there exists no domain for which Q(x) is true if and only if Q(x) is false for every domain. Let C(x) denote "x eats cheeseburgers." Then the statement "All Americans eat cheeseburgers" is represented by∀xC(x), where the domain consists of all Americans.
Translating from English into Logical Expressions
There is a student in this class with the property that the student visited Mexico.” For each x in this class, x has the property that x has visited Mexico or x has visited Canada.”.
Using Quantifiers in System Specifications
Note that the second statement cannot be written as ∃x(P (x)→ ¬R(x)). The reason is that P (x)→ ¬R(x) is true whenever it is not a lion, so that ∃x( P (x)→ ¬R(x)) is true as long as there is at least one creature that is not a lion, even if every lion drinks coffee. Note that we have assumed that "small" is the same as "not large" and that "dull color" is the same as "not richly colored."
Logic Programming
Nested Quantifiers
Although nested quantifiers can sometimes be difficult to understand, the rules we already studied in Section 1.4 can help us use them. We will see how to use nested quantifiers to express mathematical statements such as "The sum of two positive integers is always positive." We will show how nested quantifiers can be used to translate English sentences such as "Everyone has exactly one best friend" into logical statements.
Understanding Statements Involving Nested Quantifiers
Finally, to see if ∃x∃yP (x, y) is true, we loop through the values forx, where for each x we loop through the values fory until we hit anx for which we hit ay for which P (x, y) where is. The statement∃x∃yP (x, y) is false only if we never hit anx for which we hit such that P (x, y) is true.
The Order of Quantifiers
If we find that P (x, y) is true for all values for xandy, we have found that ∀x∀yP (x, y) is true. On the other hand, ∀x∃yP (x, y) is true if and only if for every value of x there is a value of y for which P (x, y) is true.
Translating Mathematical Statements into Statements Involving Nested Quantifiers
There is a real number such that for all real numbers x and for all real numbers sy, x+y = z,". Solution: We first rewrite this as "For every real number x other than zero, x has a multiplicative inverse." This can be rewritten as "For every real number x, if x=0, then there exists a real number such that xy=1." This can be rewritten as
Translating from Nested Quantifiers into English
An example that you may be familiar with is the concept of limit, which is important in calculus. It follows that the original statement, which is triple quantified, says that there is a studentx such that for all students and all other students if they are friends and other friends are not friends.
Translating English Sentences into Logical Expressions
EXAMPLE 13 Use quantifiers to express the statement "There is a woman who has flown on every airline in the world." Solution: Let P (w, f ) be "whas takenf" and Q(f, a)be "f is a flight ona." We can express the statement as.
Negating Nested Quantifiers
Rules of Inference
We will describe how we can use these rules of reasoning to make valid arguments. Finally, we will show how inference rules for propositions and for quantified statements can be combined.
Valid Arguments in Propositional Logic
After we illustrate how rules of inference are used to produce valid arguments, we will describe some common types of faulty reasoning, called fallacies, that lead to invalid arguments. After studying inference rules in propositional logic, we will introduce inference rules for quantified statements.
Rules of Inference for Propositional Logic
EXAMPLE 3 Indicate which inference rule is the basis of the following argument: “It is now below freezing. Solution: Let the statement be 'It is now below freezing' and the statement be 'It is raining now'. Then this argument is of the form.
Using Rules of Inference to Build Arguments
Solution: Let the proposition be "It is raining today", let the proposition be "We won't have a barbecue today", and let the proposition be "We will have a barbecue tomorrow". Then this argument is of the form. Solution: Let the proposition be "Send me an e-mail message", the proposition "I will finish writing the program", the proposition "I will go to bed early", and the proposition "I will wake up feeling refreshed". Then the premises arep→q,¬p→r, andr→s.
Resolution
EXAMPLE 7 Show that the premises "If you send me an e-mail message, then I will finish writing the program", "If you do not send me an e-mail message, then I will go to bed early" and " If I go to bed early, I'll wake up feeling rested" leads to the conclusion "If I don't finish writing the program, I'll wake up feeling refreshed."
Fallacies
Rules of Inference for Quantified Statements
EXAMPLE 12 Show that the premises "Everyone in this discrete math class has taken a computer science course" and "Marla is a student in this class" imply the conclusion "Marla has taken a computer science course." Solution: Let D(x) denote "xis in this discrete math class" and let C(x) denote "x has taken a course in computer science." Then the premises are ∀x(D(x)→C(x))andD(Marla).
Combining Rules of Inference for Propositions and Quantified Statements
- Introduction to Proofs
Use resolution to show that the hypotheses "It doesn't rain or Yvette doesn't have her umbrella," "Yvette doesn't have her umbrella or she doesn't get wet," and "It rains or Yvette doesn't get wet" imply that "Yvette don't get wet.”. Using these ingredients and rules of inference, the final step of the proof determines the truth of the statement being proved.
Some Terminology
In practice, proofs of theorems designed for human consumption are almost always informal proofs, where more than one inference rule may be used at each step, where steps may be omitted, where the axioms assumed and the inference rules used are not in explicitly. stated. Informal proofs can often explain to humans why theorems are true, while computers are perfectly happy producing formal proofs using automated reasoning systems.
Understanding How Theorems Are Stated
Moreover, when proving theorems of this kind, the first step of the proof usually involves choosing a general element of the field. To prove a theorem of the form ∀x(P (x)→Q(x)), our goal is to show that P (c)→Q(c) is true, where is an arbitrary element of the domain, and after we apply universal generalization.
Direct Proofs
Note that throughout this book we will always assume the axioms for real numbers found in Appendix 1. By the definition of an odd integer, we can conclude that n2 is an odd integer (it is once more than twice an integer).
Proof by Contraposition
The first step in a proof by contraposition is to assume that the conclusion of the conditional statement “If 3n+2 is odd, then it is odd” is false; namely, assume that even that is. The first step in a proof by contraposition is to assume that the conclusion of the conditional statement is “Ifn=ab, whileandable positive integers, thena≤√.
Proofs by Contradiction
Note that we can rewrite a proof by contraposition of a conditional statement as a proof by contradiction. Example 11 illustrates how a proof by contraposition of a conditional statement can be rewritten as a proof by contradiction.
Mistakes in Proofs
Then our hypothesis is¬P (n) and the statement “If is positive, then n2 is positive” is the statement ∀n(P (n)→Q(n)). From the hypothesis¬P (n) and the statement∀n(P (n)→Q(n)) we cannot infer¬Q(n), because we are not using a valid inference rule.
Just a Beginning
Proof Methods and Strategy
In this section, after developing a versatile arsenal of proof methods, we will examine some aspects of the art and science of proof. By looking at tiles like this, we will be able to quickly form conjectures and prove theorems without first developing a theory.
Exhaustive Proof and Proof by Cases
An exhaustive proof is a special type of proof in cases where each case involves checking a single example. Solution: We will use proof by contradiction, the concept without loss of generality and proof by cases.
Existence Proofs
That is, we will show that the first player always has a winning strategy without explicitly describing the moves this player should follow. Now, suppose the first player starts the game by eating only the cookie in the lower right corner.
Uniqueness Proofs
Note that we have shown that a winning strategy exists, but we have not specified the actual winning strategy. In fact, no one has been able to describe a winning strategy for that Chomp that applies to all rectangular grids by describing the moves that the first player should follow.
Proof Strategies
Show that the first player can win the game no matter what the second player does. We can reverse this argument to show that the first player can always make moves so that this player wins the game, regardless of what the second player does.
Looking for Counterexamples
Proof Strategy in Action
Tilings
This is contrary to the assumption that we can use dominoes to cover a standard checkerboard with opposite corners. EXAMPLE 22 Can we use flat triomines to cover a standard checkerboard with one of the four corners removed.
The Role of Open Problems
Wiles' quest to find a proof of Fermat's last theorem using this powerful theory, described in a program in the Nova series on public television, took nearly ten years. The interested reader should consult [Ro10] for more information about Fermat's last theorem and for additional references regarding this problem and its solution.).
Additional Proof Methods
Prove that there exists a positive integer that is equal to the sum of positive integers that do not exceed it. For each of five different tetromins, prove or disprove that you can line up a standard checkerboard with these tetromins.
Key Terms and Results
Prove that when a white square and a black square are removed from an 8×8 chessboard (colored as in the text), you can tile the remaining squares of the chessboard with dominoes. Tip: Show that when one black and one white square are removed, each part of the partition of the remaining cells, formed by inserting the barriers in the figure, can be covered by dominoes.
Review Questions
Supplementary Exercises
For which of the eight possible three-answer sequences will the student pass the test. For which of the 16 possible four-answer sequences will the student pass the test.
Computer Projects
Express the statement "There is exactly one student in this class who has taken exactly one math class at this school" using the uniqueness quantifier. Prove that there exists a positive integer that can be written as the sum of the squares of positive integers in two different ways.
Computations and Explorations
Writing Projects
Sets
In this chapter we will establish the surprising result that the set of rational numbers is countable, while the set of real numbers is not. His contributions to this field include the discovery that the set of real numbers is uncountable.
Venn Diagrams
The empty set can also be denoted by { }(that is, we represent the empty set with a pair of parentheses that enclose all the elements in this collection). The empty set can be thought of as an empty folder and the set consisting only of the empty set can be thought of as a folder with exactly one folder inside, namely the empty folder.
Subsets
A useful way to show that two sets have the same elements is to show that each set is a subset of the other. Showing that two sets are equal To show that two sets A and B are equal, show that AB and BA.
The Size of a Set
Power Sets
Note that the empty set and the set itself are members of this set of subsets.
Cartesian Products
Solution: The Cartesian product A×B consists of all the ordered pairs of the form (a,b), where is a student at the university and is a course offered at the university. EXAMPLE 18 Show that the Cartesian productB×A is not equal to the Cartesian productA×B, whereA andBaar as in Example 17.
Using Set Notation with Quantifiers
EXAMPLE 21 What are the ordered pairs in the ratio less than or equal to that contain(a, b)ifa≤b,.
Truth Sets and Quantifiers
Set Operations
EXAMPLE 4 The intersection of the set of all computer science majors at your school and the set of all math majors is the set of all students who are joint math and computer majors. EXAMPLE 9 Let be the set of positive integers greater than 10 (with universal set the set of all positive integers).
Set Identities
Using the definition of the complement of a set, we see that this implies that x ∈Aorx∈B. Proving a set identity involving more than two sets by showing each side of the identity is a subset of the other often requires us to keep track of different cases, as illustrated by the proof in Example 12 of one of the distributive laws for quantities.
Generalized Unions and Intersections
To see that the union of these sets is the set of positive integers, note that every positive integer is in at least one of the sets because it belongs to An= {1,2,. To see that the intersection of these sets is the set {1}, note that the only element that belongs to all the sets A1, A2.
Computer Representation of Sets
Functions
The range of the function we specified is the set of different ages of these students, which is the set{21,22,24}. Remark: The notation f (S) for the image of the set S under the function f is potentially ambiguous.
One-to-One and Onto Functions
Solution: The function f is one-to-one because f takes different values at the four elements. EXAMPLE 10 Determine whether the function f (x)=x+1 is one-to-one from the set of real numbers to itself.
Inverse Functions and Compositions of Functions
Solution: The function f (x)=x2 from the set of nonnegative real numbers to the set of nonnegative real numbers is one-to-one. Because the function f (x)=x2 from the set of non-negative real numbers to the set of non-negative real numbers is one-to-one and on, it is invertible.
The Graphs of Functions
Some Important Functions
Table 1, showing a real number, shows some simple but important properties of floor and ceiling functions. Each property in this table can be defined using floor and ceiling function definitions.
Partial Functions
- Sequences and Summations
Under what conditions is the function one-to-one if it assigns a student his or her a) mobile phone number. Under what conditions is the position one-to-one when assigned to a teacher.
Sequences
EXAMPLE 3 The sequences {sn}withsn= −1+4nand{tn}withn =7−3 are both arithmetic progressions with initial terms and common differences equal to -1 and 4, and 7 and -3, respectively, if we start atn =0. are often used in computer science.
Recurrence Relations
Since the initial conditions tell us that f0=0 and f1=1, using the inverse relation in the definition we find this. We say that we have solved the inverse relation together with the initial conditions when we find an explicit formula, called a closed formula, for the terms of the sequence.
Special Integer Sequences
Note that when we use iteration, we essentially guess at the formula for the sequence terms. Solution: Note that each of the first 10 terms of this sequence after the first is obtained by adding 6 to the previous term.
Summations
Cardinality of Sets
In Definition 4 of Section 2.1, we defined the cardinality of a finite set as the number of elements in the set. In Exercise 79 of Section 2.3, we showed that there is a one-to-one correspondence between any two finite sets with the same number of elements.
Countable Sets
We will find the surprising result that the set of rational numbers is countably infinite. Note that we can show that the set of all integers is countable by listing its members.
An Uncountable Set
Matrices
The set of matrix ismatrices. A matrix with the same number of rows as columns is called a square. Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in each position are equal.
Matrix Arithmetic
