Thanks to the feedback I received and the significant investment in book development, I was able to significantly improve the appeal and effectiveness of this edition of the book by the time it was released. This new edition of the book includes many improvements, updates, additions, and corrections, all designed to make the book a more effective teaching tool for the modern discrete mathematics course.
Features of the Book
These exercises are generally more difficult than those in the exercise sets after the sections. All are designed to expose students to ideas not covered in depth in the text.
How to Use This Book
These suggested readings include books at or below the level of this text, more difficult books, expository articles, and articles in which discoveries in discrete mathematics were originally published. This suggested reading is complemented by the many links to valuable resources available on the Internet and found on this book's website.
Ancillaries
Acknowledgments
The Online Learning Center
- The Information Center
- Student Site
- Instructor Site
- Connect
SmartBook Connect also offers another enhanced online version of the text on the McGraw-Hill SmartBook platform. Second, discrete mathematics is the gateway to more advanced courses in all parts of the mathematical sciences.
Propositional Logic
- Introduction
- Propositions
- Conditional Statements
- Truth Tables of Compound Propositions
- Precedence of Logical Operators
- Logic and Bit Operations
The proposal reads "notp." The truth value of the negation ofp,¬p, is the opposite of the truth value ofp. What is the value of the variablex after the statement if2+2=4thenx:=x+1. ifx=0 before this statement is encountered.
Exercises
Applications of Propositional Logic
- Introduction
- Translating English Sentences
- System Specifications
- Boolean Searches
- Logic Puzzles
- Logic Circuits
EXAMPLE 3 Express the specification "The automatic response cannot be sent when the file system is full". EXAMPLE 5 Does the system specification in Example 4 remain consistent if the specification "The diagnostic message is not retransmitted" is added.
Propositional Equivalences
- Introduction
- Logical Equivalences
- Using De Morgan’s Laws
- Constructing New Logical Equivalences
- Satisfiability
- Applications of Satisfiability
- Solving Satisfiability Problems
Since the truth values of the compound propositions ¬(p∨q) and¬p∧ ¬q match 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 are logically equivalent. 72. Explain the steps in the construction of the compound proposition given in the text that asserts that each of the nine 3 × 3 blocks of a 9 × 9 sudoku contains every number.
Predicates and Quantifiers
- Introduction
- Predicates
- Quantifiers
- QUANTIFIERS OVER FINITE DOMAINS
- Quantifiers with Restricted Domains
- Precedence of Quantifiers
- Binding Variables
- Logical Equivalences Involving Quantifiers
- Negating Quantified Expressions
- Translating from English into Logical Expressions
- Using Quantifiers in System Specifications
- Examples from Lewis Carroll
- Logic Programming
EXAMPLE 8 Let P(x) be the statement “x+1>x”. What is the truth value of the quantification ∀xP(x), where the domain consists of all real numbers. A statement∀xP(x) is false, where P(x) is a propositional function, if and only ifP(x) is not always true whenxis in the domain. EXAMPLE 9 Let Q(x) be the statement “x<2”. What is the truth value of the quantification ∀xQ(x), where the domain consists of all real numbers.
What is the truth value of this statement if the domain consists of all integers. EXAMPLE 14 Let Q(x) denote the statement “x=x+1”. What is the truth value of the quantification∃xQ(x), where the domain consists of all real numbers.
Nested Quantifiers
- Introduction
- Understanding Statements Involving Nested Quantifiers
- The Order of Quantifiers
- Translating Mathematical Statements into Statements Involving Nested Quantifiers
- Translating from Nested Quantifiers into English
- Translating English Sentences into Logical Expressions
- Negating Nested Quantifiers
Express each of these statements using quantifiers; logical connections; and P(x), Q(x), and R(x), where the domain consists of all people. EXAMPLE 2 Translate the statement into English. where the domain for both variables consists of all real numbers. The statement∃y∀xP(x, y) is true if and only if there is something that makes P(x, y) true for every x.
Let L(x, y) be the statement "xlovesy", where the domain for dyxandy consists of all the people in the world. There is a student in your class who has been chatting with everyone in your class online.
Rules of Inference
- Introduction
- Valid Arguments in Propositional Logic
- Rules of Inference for Propositional Logic
- Using Rules of Inference to Build Arguments
- Resolution
- Fallacies
- Rules of Inference for Quantified Statements
- Combining Rules of Inference for Propositions and Quantified Statements
We then show that the resulting form of the argument is the inference rule from Table 1. EXAMPLE 3 State which inference rule is the basis of the following argument: “It's below freezing now. EXAMPLE 4 State which rule of reasoning underlies the following argument: “It is below freezing and raining now. When there are many premises, several rules of reasoning are often needed to show that the argument is valid.
We will illustrate how some of these inference rules for quantified statements are applied in Examples 12 and 13. Explain the inference rules used to obtain each conclusion from the premises. a) "If I take a day off, it rains or snows."
Justify the rule of universal transitivity, which states that if ∀x(P(x) → Q(x)) and ∀x(Q(x) → R(x)) are true,
- Introduction to Proofs
- Introduction
- Some Terminology
- Understanding How Theorems Are Stated
- Methods of Proving Theorems
- Direct Proofs
- Proof by Contraposition
- Proofs by Contradiction
- Mistakes in Proofs
- Just a Beginning
- Proof Methods and Strategy
- Introduction
- Exhaustive Proof and Proof by Cases
- Existence Proofs
- Uniqueness Proofs
- Proof Strategies
- Looking for Counterexamples
- Proof Strategy in Action
- Tilings
- The Role of Open Problems
We illustrate the use of counterexamples in Example 15. EXAMPLE 15 Show that the statement "Every positive integer is the sum of the squares of two integers" is false. That is, the statement "Every positive integer is the sum of the squares of three integers" is true or false. We conclude that the statement "Every positive integer is the sum of the squares of three integers" is false.
We have shown that not every positive integer is the sum of the squares of three integers. It turns out that the conjecture "Every positive integer is the sum of the squares of four integers" is true.
- Additional Proof Methods
Prove that there are no positive perfect cubes less than 1000 that are the sum of the cubes of two positive integers. Prove that there exists a positive integer that is equal to the sum of positive integers that do not exceed it. Show that if,b, and care real numbers anda≠0, then there is a unique solution to the equation tax+b=c.
Prove that when a white square and a black square are removed from an 8×8 checkerboard (colored as in the text), you can tile the remaining squares of the board with dominoes. For each of the five different tetrominoes, prove or disprove that you can tile a standard checkerboard using those tetrominoes.
Key Terms and Results
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. Show that by removing two white squares and two black squares from an 8×8 chessboard (colored as in the text) you can make it impossible to tile the remaining squares with dominoes. Find all the squares, if they exist, on an 8×8 chessboard, so that the board obtained by removing one of these squares can be tiled with straight triominoes.
Tip: First use arguments based on coloring and rotations to remove as many squares as possible from consideration.].
Review Questions
Use rules of inference to show that if the premises "All zebras have stripes" and "Mark is a zebra" are true, then the conclusion "Mark has stripes" is true. To prove that the statements p1,p2,p3 and p4 are equivalent, is it sufficient to show that the conditional statements p4→p2,p3→p1, andp1→p2 are valid? If not, provide another set of conditional statements that can be used to show that the four statements are equivalent.
What are the elements of a proof that there is a unique elementxsuch that P(x), where P(x) is a propositional function. Explain how a proof by case can be used to prove a result about absolute values, such as the fact that|xy|=|x||y| for all real numbersxandy.
Supplementary Exercises
Which of the eight possible sets of three answers will the student pass the test for? Which of the 16 possible sets of four answers will the student pass the test for? Show that the argument with premises "The Tooth Fairy is a real person" and "The Tooth Fairy is not a real person" and the conclusion "You can find gold at the end of the rainbow" is a valid argument.
Suppose the truth value of the proposition pi is T if i is an odd positive integer, and is F if it is an even positive integer. Disprove the claim that every positive integer is the sum of the cubes of eight non-negative integers.
Computer Projects
Express this statement using quantifiers: "There is a building on the campus of some college in the United States in which every room is painted white." Express the statement: "There is exactly one student in this class who has exactly took one math class at this school,” using the uniqueness quantifier. Describe an inference rule that can be used to prove that there are exactly two elements xandy in a domain such that P(x) and P(y) are true.
Prove that there exists a positive integer that can be written as the sum of the squares of positive integers in two different ways. Assuming that the theorem stating that √ is irrational whenever there is a positive integer that is not a perfect square, prove that √.
Computations and Explorations
Express this statement using quantifiers: "Every student in this class has taken a course in every department of the School of Mathematical Sciences." Prove the statement that every positive integer is the sum of at most two squares and a cube of non-negative integers. Prove the statement that every positive integer is the sum of 36 fifth powers of non-negative integers.
Writing Projects
Sets
- Introduction
- Venn Diagrams
- Subsets
- The Size of a Set
- Power Sets
- Cartesian Products
- Using Set Notation with Quantifiers
- Truth Sets and Quantifiers
His contributions in this field include the discovery that the set of real numbers is uncountable. For example, the set of all positive integers greater than their squares is the null set. Solution: We draw a rectangle to represent the universal setU, which is the collection of the 26 letters of the English alphabet.
Note that the empty set and the set itself are members of this set of subsets. Because|x|=x if and only ifx≥0, it follows that the truth set of RisN, the set of non-negative integers.
Set Operations
- Introduction
- Set Identities
- Generalized Unions and Intersections
- Computer Representation of Sets
- Multisets
This identity states that the complement of the intersection of two sets is the union of their complements. Using the definition of the complement of a set, we see that this implies that x∈Aorx∈B. Solution: We will prove this identity by showing that each side is a subset of the other side.
Subset Method Show that each side of the identity is a subset of the other side. Find the symmetric difference between the set of computer science majors at a school and the set of math majors at that school.
Functions
- Introduction
- One-to-One and Onto Functions
- Inverse Functions and Compositions of Functions
- The Graphs of Functions
- Some Important Functions
- Partial Functions
The range of the function we have specified is the set of different ages of these students, namely the set {21,22,24}. Note: We can express that f is one-to-one using quantifiers as∀a∀b(f(a)=f(b)→a=b) or equivalently∀a∀b(a≠b→f( a)≠ f(b)), where the universe of the discourse is the domain of the function. EXAMPLE 26 Graph the function f(n)=2n+1 from the set of integers to the set of integers.
Solution: The off graph is the set of ordered pairs of the form (n,2n+1), where is an integer. EXAMPLE 27 Plot the graph of the function tf(x)=x2 from the set of integers to the set of integers.
- Sequences and Summations
- Introduction
- Sequences
- Recurrence Relations
- Special Integer Sequences
- Summations
The terms of a series can be specified by specifying a formula for each term of the series. We describe sequences by listing the terms of the sequence in order of ascending subscripts. We say that we have solved the recurrence relation together with the initial conditions when we find an explicit formula, a so-called closed formula, for the terms of the series.
At each iteration of the recurrence relation, we get the next term in the sequence by adding 3 to the previous term. Note that when we use iteration, we are essentially guessing a formula for the terms of the sequence.
