Bond Robert J., Keane William J. An Introduction to Abstract Mathematics

Waveland Pr Inc., 2007. — 323 р.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.Mathematical Reasoning
Statements
Compound Statements
Implications
Contrapositive and Converse
Sets
Sets and Subsets
Combining Sets
Collections of Sets
Functions
Definition and Basic Properties
Surjective and Injective Functions
Composition and Invertible Functions
Binary Operations and Relations
Binary Operations
Equivalence Relations
The Integers
Axioms and Basic Properties
Induction
The Division Algorithm and Greatest Common Divisors
Primes and Unique Factorization
Congruences
Generalizing a Theorem
Infinite Sets
Countable Sets
Uncountable Sets, Cantor's Theorem, and the Schroeder-Bernstein Theorem
Collections of Sets
The Real and Complex Numbers
Fields
The Real Numbers
The Complex Numbers
Polynomials
Polynomials
Unique Factorization
Polynomials over C, R, and Q
Answers and Hints to Selected Exercises