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El. knyga: Introduction to Mathematical Structures and Proofs

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As a student moves from basic calculus courses into upper-division courses in linear and abstract algebra, real and complex analysis, number theory, topology, and so on, a "bridge" course can help ensure a smooth transition. Introduction to Mathematical Structures and Proofs is a textbook intended for such a course, or for self-study.  This book introduces an array of fundamental mathematical structures. It also explores the delicate balance of intuition and rigorand the flexible thinkingrequired to prove a nontrivial result.  In short, this book seeks to enhance the mathematical maturity of the reader.

The new material in this second edition includes a section on graph theory, several new sections on number theory (including primitive roots, with an application to card-shuffling), and a brief introduction to the complex numbers (including a section on the arithmetic of the Gaussian integers). Solutions for even numbered exercises are available on springer.com forinstructors adopting the text for a course.
Preface to the Second Edition v
Preface to the First Edition vii
1 Logic
1(36)
1.1 Statements, Propositions, and Theorems
1(5)
1.2 Logical Connectives and Truth Tables
6(6)
1.3 Conditional Statements
12(6)
1.4 Proofs: Structures and Strategies
18(8)
1.5 Logical Equivalence
26(5)
1.6 Application: A Brief Introduction to Switching Circuits
31(6)
2 Sets
37(72)
2.1 Fundamentals
37(8)
2.2 Russell's Paradox
45(1)
2.3 Quantifiers
46(7)
2.4 Set Inclusion
53(5)
2.5 Union, Intersection, and Complement
58(5)
2.6 Indexed Sets
63(8)
2.7 The Power Set
71(4)
2.8 Ordered Pairs and Cartesian Products
75(7)
2.9 Set Decomposition: Partitions and Relations
82(15)
2.10 Mathematical Induction and Recursion
97(12)
3 Functions
109(32)
3.1 Definitions and Examples
109(9)
3.2 Surjections, Injections, Bijections, Sequences
118(13)
3.3 Composition of Functions
131(10)
4 Finite and Infinite Sets
141(50)
4.1 Cardinality; Fundamental Counting Principles
141(17)
4.2 Comparing Sets, Finite or Infinite
158(7)
4.3 Countable and Uncountable Sets
165(10)
4.4 More on Infinity
175(1)
4.5 Languages and Finite Automata
176(15)
5 Combinatorics
191(86)
5.1 Combinatorial Problems
191(3)
5.2 The Addition and Product Rules (review)
194(2)
5.3 Introduction to Permutations
196(11)
5.4 Permutations and Geometric Symmetry
207(7)
5.5 Decomposition into Cycles
214(12)
5.6 The Order of a Permutation; A Card-Shuffling Example
226(7)
5.7 Odd and Even Permutations; Applications to Configurations
233(10)
5.8 Binomial and Multinomial Coefficients
243(18)
5.9 Graphs
261(16)
6 Number Theory
277(72)
6.1 Operations
278(6)
6.2 The Integers: Operations and Order
284(3)
6.3 Divisibility: The Fundamental Theorem of Arithmetic
287(15)
6.4 Congruence; Divisibility Tests
302(7)
6.5 Introduction to Euler's Function
309(6)
6.6 The Inclusion-Exclusion Principle and Euler's Function
315(7)
6.7 More on Prime Numbers
322(5)
6.8 Primitive Roots and Card Shuffling
327(9)
6.9 Perfect Numbers, Mersenne Primes, Arithmetic Functions
336(10)
6.10 Number Theory and Cryptography: A Brief Glimpse
346(3)
7 Complex Numbers
349(26)
7.1 Complex Numbers
349(13)
7.2 The Gaussian Integers
362(13)
Hints and Partial Solutions to Selected Odd-Numbered Exercises 375(20)
Index 395
Larry Gerstein's primary areas of research have been in quadratic forms and number theory and he has published extensively in these areas. The author's first edition of "Introduction to Mathematical Structures and Proofs" has sold to date (8/2/2010) over 6000 copies and has gone through 5 printings. Gerstein himself has a transition course at UC, Santa Barbara (Math 8-A transition to higher mathematics) from his book since its first publication date. The first edition also received 2 glowing reviews by Steve Krantz for the American Mathematical Monthly, and S. Gottwald for Zentralblatt.
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