An Introduction to Abstract Mathematics

✍ Scribed by Robert J. Bond, William J. Keane

• Publisher: Waveland Pr Inc
• Year: 2007
• Tongue: English
• Leaves: 345
• Category: Library

✦ Synopsis

Bond and Keane explicate the elements of logical, mathematical argument to elucidate the meaning and importance of mathematical rigor. With definitions of concepts at their disposal, students learn the rules of logical inference, read and understand proofs of theorems, and write their own proofs--all while becoming familiar with the grammar of mathematics and its style. In addition, they will develop an appreciation of the different methods of proof (contradiction, induction), the value of a proof, and the beauty of an elegant argument. The authors emphasize that mathematics is an ongoing, vibrant discipline--its long, fascinating history continually intersects with territory still uncharted and questions still in need of answers. The authors' extensive background in teaching mathematics shines through in this balanced, explicit, and engaging text, designed as a primer for higher-level mathematics courses. They elegantly demonstrate process and application and recognize the byproducts of both the achievements and the missteps of past thinkers. Chapters 1-5 introduce the fundamentals of abstract mathematics and chapters 6-8 apply the ideas and techniques, placing the earlier material in a real context. Readers' interest is continually piqued by the use of clear explanations, practical examples, discussion and discovery exercises, and historical comments.

✦ Table of Contents

Mathematical Reasoning 1
1.1 Statements 2
1.2 Compound Statements 16
1.3 Implications 29
1.4 Contrapositive and Converse 38

Sets 49
2.1 Sets and Subsets 49
2.2 Combining Sets 61
2.3 Collections of Sets 72

Functions 81
3.1 Definition and Basic Properties 81
3.2 Surjective and Injective Functions 97
3.3 Composition and Invertible Functions 110

Binary Operations and Relations 123
4.1 Binary Operations 123
4.2 Equivalence Relations 139

The Integers 151
5.1 Axioms and Basic Properties 151
5.2 Induction 159
5.3 The Division Algorithm and Greatest Common Divisors 175
5.4 Primes and Unique Factorization 182
5.5 Congruences 189
5.6 Generalizing a Theorem 200

Infinite Sets 209
6.1 Countable Sets 210
6.2 Uncountable Sets, Cantor's Theorem, and the Schroeder-Bernstein Theorem 220
6.3 Collections of Sets 229

The Real and Complex Numbers 235
7.1 Fields 235
7.2 The Real Numbers 243
7.3 The Complex Numbers 251

Polynomials 263
8.1 Polynomials 263
8.2 Unique Factorization 273
8.3 Polynomials over C, R, and Q 285

Answers and Hints to Selected Exercises 295
Bibliography 317
Index 319

✦ Subjects

Математика;Элементарная математика;