Mathematical Proofs: A Transition to Advanced Mathematics

by Chartrand, Gary; Polimeni, Albert D.; Zhang, Ping

• ISBN13: 9780201710908
• ISBN10: 0201710900
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-01-01
• Publisher: Addison Wesley

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