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Mathematical Proofs: A Transition to Advanced Mathematics, 2/e,prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.KEY TOPICS: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory.MARKET: For all readers interested in advanced mathematics and logic.
0. Communicating Mathematics Learning Mathematics What Others Have Said About Writing Mathematical Writing Using Symbols Writing Mathematical Expressions Common Words and Phrases in Mathematics Some Closing Comments about Writing
1. Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets 1.6 Cartesian Products of Sets Exercises for Chapter 1
2. Logic 2.1 Statements 2.2 The Negation of a Statement 2.3 The Disjunction and Conjunction of Statements 2.4 The Implication 2.5 More on Implications 2.6 The Biconditional 2.7 Tautologies and Contradictions 2.8 Logical Equivalence 2.9 Some Fundamental Properties of Logical Equivalence 2.10 Quantified Statements 2.11 Characterizations of Statements Exercises for Chapter 2
3. Direct Proof and Proof by Contrapositive 3.1 Trivial and Vacuous Proofs 3.2 Direct Proofs 3.3 Proof by Contrapositive 3.4 Proof by Cases 3.5 Proof Evaluations Exercises for Chapter 3
4. More on Direct Proof and Proof by Contrapositive 4.1 Proofs Involving Divisibility of Integers 4.2 Proofs Involving Congruence of Integers 4.3 Proofs Involving Real Numbers 4.4 Proofs Involving Sets 4.5 Fundamental Properties of Set Operations 4.6 Proofs Involving Cartesian Products of Sets Exercises for Chapter 4
5. Existence and Proof by Contradiction 5.1 Counterexamples 5.2 Proof by Contradiction 5.3 A Review of Three Proof Techniques 5.4 Existence Proofs 5.5 Disproving Existence Statements Exercises for Chapter 5
6. Mathematical Induction 6.1 The Principle of Mathematical Induction 6.2 A More General Principle of Mathematical Induction 6.3 Proof by Minimum Counterexample 6.4 The Strong Principle of Mathematical Induction Exercises for Chapter 6
7. Prove or Disprove 7.1 Conjectures in Mathematics 7.2 Revisiting Quantified Statements 7.3 Testing Statements 7.4 A Quiz of "Prove or Disprove" Problems Exercises for Chapter 7 8. Equivalence Relations
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Mathematical Proofs: A Transition to Advanced Mathematics, 2/e,prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.KEY TOPICS: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory.MARKET: For all readers interested in advanced mathematics and logic.
Table of Contents
0. Communicating Mathematics Learning Mathematics What Others Have Said About Writing Mathematical Writing Using Symbols Writing Mathematical Expressions Common Words and Phrases in Mathematics Some Closing Comments about Writing
1. Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets 1.6 Cartesian Products of Sets Exercises for Chapter 1
2. Logic 2.1 Statements 2.2 The Negation of a Statement 2.3 The Disjunction and Conjunction of Statements 2.4 The Implication 2.5 More on Implications 2.6 The Biconditional 2.7 Tautologies and Contradictions 2.8 Logical Equivalence 2.9 Some Fundamental Properties of Logical Equivalence 2.10 Quantified Statements 2.11 Characterizations of Statements Exercises for Chapter 2
3. Direct Proof and Proof by Contrapositive 3.1 Trivial and Vacuous Proofs 3.2 Direct Proofs 3.3 Proof by Contrapositive 3.4 Proof by Cases 3.5 Proof Evaluations Exercises for Chapter 3
4. More on Direct Proof and Proof by Contrapositive 4.1 Proofs Involving Divisibility of Integers 4.2 Proofs Involving Congruence of Integers 4.3 Proofs Involving Real Numbers 4.4 Proofs Involving Sets 4.5 Fundamental Properties of Set Operations 4.6 Proofs Involving Cartesian Products of Sets Exercises for Chapter 4
5. Existence and Proof by Contradiction 5.1 Counterexamples 5.2 Proof by Contradiction 5.3 A Review of Three Proof Techniques 5.4 Existence Proofs 5.5 Disproving Existence Statements Exercises for Chapter 5
6. Mathematical Induction 6.1 The Principle of Mathematical Induction 6.2 A More General Principle of Mathematical Induction 6.3 Proof by Minimum Counterexample 6.4 The Strong Principle of Mathematical Induction Exercises for Chapter 6
7. Prove or Disprove 7.1 Conjectures in Mathematics 7.2 Revisiting Quantified Statements 7.3 Testing Statements 7.4 A Quiz of "Prove or Disprove" Problems Exercises for Chapter 7 8. Equivalence Relations
Digital Rights
eTextbooks and eChapters can be viewed by using the free reader listed below.
Be sure to check the format of the eTextbook/eChapter you purchase to know which reader you will need. After purchasing your eTextbook or eChapter, you will be emailed instructions on where and how to download your free reader.
Download Requirements:Due to the size of eTextbooks, a high-speed Internet connection (cable modem, DSL, LAN) is required for download stability and speed. Your connection can be wired or wireless.
Being online is not required for reading an eTextbook after successfully downloading it. You must only be connected to the Internet during the download process.
User Help:
Click Here to access the VitalSource Bookshelf FAQ
Digital Rights Management (DRM) Key
Printing - Books that cannot be printed will show "Not Allowed." Otherwise, this will detail the number of times it can be printed, or "Allowed with no limits."
Expires - Books that have no expiration (the date upon which you will no longer be able to access your eBook) will read "No Expiration." Otherwise it will state the number of days from activation (the first time you actually read it).
Reading Aloud - Books enabled with the "text-to-speech" feature so that they can be read aloud will show "Allowed."
Sharing - Books that cannot be shared with other computers will show "Not Allowed."
Min. Software Version - This is the minimum software version needed to read this book.
Suitable Devices - Hardware known to be compatible with this book. Note: Reader software still needs to be installed.