
Mathematical Proofs: A Transition to Advanced Mathematics
Description
Alles über E-Books | Antworten auf Fragen rund um E-Books, Kopierschutz und Dateiformate finden Sie in unserem Info- & Hilfebereich.
All prices
More details
Other editions
Additional editions

Content
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
Exercises for Chapter 7
8. Equivalence Relations
8.1 Relations
8.2 Properties of Relations
8.3 Equivalence Relations
8.4 Properties of Equivalence Classes
8.5 Congruence Modulo n
8.6 The Integers Modulo n
Exercises for Chapter 8
9. Functions
9.1 The Definition of Function
9.2 The Set of All Functions from A to B
9.3 One-to-one and Onto Functions
9.4 Bijective Functions
9.5 Composition of Functions
9.6 Inverse Functions
9.7 Permutations
Exercises for Chapter 9
10. Cardinalities of Sets
10.1 Numerically Equivalent Sets
10.2 Denumerable Sets
10.3 Uncountable Sets
10.4 Comparing Cardinalities of Sets
10.5 The Schroeder-Bernstein Theorem
Exercises for Chapter 10
11. Proofs in Number Theory
11.1 Divisibility Properties of Integers
11.2 The Division Algorithm
11.3 Greatest Common Divisors
11.4 The Euclidean Algorithm
11.5 Relatively Prime Integers
11.6 The Fundamental Theorem of Arithmetic
11.7 Concepts Involving Sums of Divisors
Exercises for Chapter 11
12. Proofs in Calculus
12.1 Limits of Sequences
12.2 Infinite Series
12.3 Limits of Functions
12.4 Fundamental Properties of Limits of Functions
12.5 Continuity
12.6 Differentiability
Exercises for Chapter 12
13. Proofs in Group Theory
1
System requirements
File format: PDF
Copy-Protection: Adobe-DRM (Digital Rights Management)
System requirements:
- Computer (Windows; MacOS X; Linux): Install the free reader Adobe Digital Editions prior to download (see eBook Help).
- Tablet/smartphone (Android; iOS): Install the free app Adobe Digital Editions or the app PocketBook before downloading (see eBook Help).
- E-reader: Bookeen, Kobo, Pocketbook, Sony, Tolino and many more (only limited: Kindle).
The file format PDF always displays a book page identically on any hardware. This makes PDF suitable for complex layouts such as those used in textbooks and reference books (images, tables, columns, footnotes). Unfortunately, on the small screens of e-readers or smartphones, PDFs are rather annoying, requiring too much scrolling.
This eBook uses Adobe-DRM, a „hard” copy protection. If the necessary requirements are not met, unfortunately you will not be able to open the eBook. You will therefore need to prepare your reading hardware before downloading.
Please note: We strongly recommend that you authorise using your personal Adobe ID after installation of any reading software.
For more information, see our eBook Help page.
File format: PDF
Copy protection: Watermark-DRM (Digital Rights Management)
System requirements:
- Computer (Windows; MacOS X; Linux): Use the free software Adobe Reader, Adobe Digital Editions, or any other PDF viewer of your choice (see eBook Help).
- Tablet/Smartphone (Android; iOS): Install the free app Adobe Digital Editions or another reading app for eBooks, e.g., PocketBook (see eBook Help).
- E-reader: Bookeen, Kobo, Pocketbook, Sony, Tolino and many more (only limited: Kindle).
The file format PDF always displays a book page identically on any hardware. This makes PDF suitable for complex layouts such as those used in textbooks and reference books (images, tables, columns, footnotes). Unfortunately, on the small screens of e-readers or smartphones, PDFs are rather annoying, requiring too much scrolling.
This eBook uses Watermark-DRM, a „soft” copy protection. This means that there are no technical restrictions to prevent illegal distribution. However, there is a personalised watermark embedded in the eBook that can be used to identify the purchaser of the eBook in the event of misuse and to provide evidence for legal purposes.
For more information, see our eBook Help page.