Mathematical Proofs A Transition to Advanced Mathematics

ISBN-10: 0201710900

ISBN-13: 9780201710908

Edition: 2003

Authors: Gary Chartrand, Albert D. Polimeni, Ping Zhang

List price: $126.67
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Book details

List price: $126.67
Copyright year: 2003
Publisher: Addison-Wesley Longman, Incorporated
Publication date: 5/28/2002
Binding: Hardcover
Pages: 384
Size: 7.75" wide x 9.50" long x 0.50" tall
Weight: 1.430
Language: English

Communicating Mathematics
Learning Mathematics
What Others Have Said About Writing
Mathematical Writing
Using Symbols
Writing Mathematical Expressions
Common Words and Phrases in Mathematics
Some Closing Comments About Writing
Describing a Set
Special Sets
Set Operations
Indexed Collections of Sets
Partitions of Sets
Cartesian Products of Sets
The Negation of a Statement
The Disjunction and Conjunction of Statements
The Implication
More On Implications
The Biconditional
Tautologies and Contradictions
Logical Equivalence
Some Fundamental Properties of Logical Equivalence
Characterizations of Statements
Quantified Statements and Their Negations
Direct Proof and Proof by Contrapositive
Trivial and Vacuous Proofs
Direct Proofs
Proof by Contrapositive
Proof by Cases
Proof Evaluations
More on Direct Proof and Proof by Contrapositive
Proofs Involving Divisibility of Integers
Proofs Involving Congruence of Integers
Proofs Involving Real Numbers
Proofs Involving Sets
Fundamental Properties of Set Operations
Proofs Involving Cartesian Products of Sets
Proof by Contradiction
Proof by Contradiction
Examples of Proof by Contradiction
The Three Prisoners Problem
Other Examples of Proof by Contradiction
The Irrationality of �2
A Review of the Three Proof Techniques
Prove or Disprove
Conjectures in Mathematics
A Review of Quantifiers
Existence Proofs
A Review of Negations of Quantified Statements
Disproving Statements
Testing Statements
A Quiz of “Prove or Disprove” Problems
Equivalence Relations
Reflexive, Symmetric, and Transitive Relations
Equivalence Relations
Properties of Equivalence Classes
Congruence Modulo n
The Integers Modulo n
The Definition of function
The Set of All Functions From A to B
One-to-one and Onto Functions
Bijective Functions
Composition of Functions
Inverse Functions
Mathematical Induction
The Well-Ordering Principle
The Principle of Mathematical Induction
Mathematical Induction and Sums of Numbers
Mathematical Induction and Inequalities
Mathematical Induction and Divisibility
Other Examples of Induction Proofs
Proof By Minimum Counterexample
The Strong Form of Induction
Cardinalities of Sets
Numerically Equivalent Sets
Denumerable Sets
Uncountable Sets
Comparing Cardinalities of Sets
The Schroder-Bernstein Theorem
Proofs in Number Theory
Divisibility Properties of Integers
The Division Algorithm
Greatest Common Divisors
The Euclidean Algorithm
Relatively Prime Integers
The Fundamental Theorem of Arithmetic
Concepts Involving Sums of Divisors
Proofs in Calculus
Limits of Sequences
Infinite Series
Limits of Functions
Fundamental Properties of Limits of Functions
Proofs in Group Theory
Binary Operations
Permutation Groups
Fundamental Properties of Groups
Isomorphic Groups
Answers and Hints to Selected Odd-Numbered Exercises
References Index of Symbols
Index of Mathematical Terms
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