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Mathematical Proofs : A Transition to Advanced Mathematics,9780201710908
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Mathematical Proofs : A Transition to Advanced Mathematics

by ; ;
Edition:
2nd
ISBN13:

9780201710908

ISBN10:
0201710900
Format:
Hardcover
Pub. Date:
1/1/2008
Publisher(s):
Addison Wesley
List Price: $124.00
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Summary

Mathematical Proofs is designed to prepare 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.

Table of Contents

Communicating Mathematics
01(12)
Learning Mathematics
01(2)
What Others Have Said About Writing
03(1)
Mathematical Writing
04(1)
Using Symbols
05(2)
Writing Mathematical Expressions
07(1)
Common Words and Phrases in Mathematics
08(3)
Some Closing Comments About Writing
11(2)
Sets
13(16)
Describing a Set
13(2)
Special Sets
15(1)
Subsets
16(2)
Set Operations
18(3)
Indexed Collections of Sets
21(2)
Partitions of Sets
23(1)
Cartesian Products of Sets
24(5)
Exercises for Chapter 1
24(5)
Logic
29(22)
Statements
29(2)
The Negation of a Statement
31(1)
The Disjunction and Conjunction of Statements
32(1)
The Implication
33(2)
More On Implications
35(1)
The Biconditional
36(2)
Tautologies and Contradictions
38(1)
Logical Equivalence
39(2)
Some Fundamental Properties of Logical Equivalence
41(1)
Characterizations of Statements
42(2)
Quantified Statements and Their Negatiors
44(7)
Exercises for Chapter 2
46(5)
Direct Proof and Proof By Contrapositive
51(16)
Trivial and Vacuous Proofs
51(2)
Direct Proofs
53(3)
Proof by Contrapositive
56(4)
Proof by Cases
60(3)
Proof Evaluations
63(4)
Exercises for Chapter 3
64(3)
More on Direct Proof and Proof by Contrapositive
67(16)
Proofs Involving Divisibility of Integers
67(3)
Proofs Involving Congruence of Integers
70(3)
Proofs Involving Real Numbers
73(1)
Proofs Involving Sets
74(3)
Fundamental Properties of Set Operations
77(2)
Proofs Involving Cartesian Products of Sets
79(4)
Exercises for Chapter 4
80(3)
Proof by Contradiction
83(10)
Proof by Contradiction
83(1)
Examples of Proof by Contradiction
84(1)
The Three Prisoners Problem
85(2)
Other Examples of Proof by Contradiction
87(1)
The Irrationality of √2
87(1)
A Review of the Three Proof Techniques
88(5)
Exercises for Chapter S
90(3)
Prove or Disprove
93(20)
Conjectures in Mathematics
93(3)
A Review of Quantifiers
96(2)
Existence Proofs
98(2)
A Review of Negations of Quantified Statements
100(1)
Counterexamples
101(2)
Disproving Statements
103(2)
Testing Statements
105(2)
A Quiz of ``Prove or Disprove'' Problems
107(6)
Exercises for Chapter 6
108(5)
Equivalence Relations
113(22)
Relations
113(1)
Reflexive, Symmetric, and Transitive Relations
114(2)
Equivalence Relations
116(3)
Properties of Equivalence Classes
119(4)
Congruence Modulo n
123(4)
The Integers Modulo n
127(8)
Exercises for Chapter 7
130(5)
Functions
135(18)
The Definition of Function
135(3)
The Set of All Functions From A to B
138(1)
One-to-one and Onto Functions
138(2)
Bijective Functions
140(3)
Composition of Functions
143(3)
Inverse Functions
146(3)
Permutations
149(4)
Exercises for Chapter 8
150(3)
Mathematical Induction
153(22)
The Well-Ordering Principle
153(2)
The Principle of Mathematical Induction
155(3)
Mathematical Induction and Sums of Numbers
158(4)
Mathematical Induction and Inequalities
162(1)
Mathematical Induction and Divisibility
163(2)
Other Examples of Induction Proofs
165(1)
Proof By Minimum Counterexample
166(2)
The Strong Form of Induction
168(7)
Exercises for Chapter 9
171(4)
Cardinalities of Sets
175(22)
Numerically Equivalent Sets
176(1)
Denumerable Sets
177(6)
Uncountable Sets
183(5)
Comparing Cardinalities of Sets
188(3)
The Schroder-Bernstein Theorem
191(6)
Exercises for Chapter 10
194(3)
Proofs in Number Theory
197(18)
Divisibility Properties of Integers
197(1)
The Division Algorithm
198(4)
Greatest Common Divisors
202(2)
The Euclidean Algorithm
204(2)
Relatively Prime Integers
206(2)
The Fundamental Theorem of Arithmetic
208(2)
Concepts Involving Sums of Divisors
210(5)
Exercises for Chapter 11
211(4)
Proofs in Calculus
215(28)
Limits of Sequences
215(5)
Infinite Series
220(4)
Limits of Functions
224(6)
Fundamental Properties of Limits of Functions
230(5)
Continuity
235(2)
Differentiability
237(6)
Exercises for Chapter 12
239(4)
Proofs in Group Theory
243(26)
Binary Operations
243(4)
Groups
247(5)
Permutation Groups
252(3)
Fundamental Properties of Groups
255(2)
Subgroups
257(3)
Isomorphic Groups
260(9)
Exercises for Chapter 13
263(6)
Answers and Hints to Selected Odd-Numbered Exercises 269(12)
References 281(2)
Index of Symbols 283(2)
Index of Mathematical Terms 285


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