Mathematical Proofs A Transition to Advanced Mathematics

ISBN-10: 0321390539
ISBN-13: 9780321390530
Edition: 2nd 2008
List price: $138.67
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Description: Key Message: Mathematical Proofs: A Transition to Advanced Mathematics, Second Edition, prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their  More...

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Book details

List price: $138.67
Edition: 2nd
Copyright year: 2008
Publisher: Addison Wesley
Publication date: 10/3/2007
Binding: Hardcover
Pages: 384
Size: 8.00" wide x 9.75" long x 1.00" tall
Weight: 1.870
Language: English

Key Message: Mathematical Proofs: A Transition to Advanced Mathematics, Second Edition, prepares 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. Key Topics: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory Market: For all readers interested in advanced mathematics and logic.

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
Sets
Describing a Set
Special Sets
Subsets
Set Operations
Indexed Collections of Sets
Partitions of Sets
Cartesian Products of Sets
Logic
Statements
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
Counterexamples
Disproving Statements
Testing Statements
A Quiz of “Prove or Disprove” Problems
Equivalence Relations
Relations
Reflexive, Symmetric, and Transitive Relations
Equivalence Relations
Properties of Equivalence Classes
Congruence Modulo n
The Integers Modulo n
Functions
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
Permutations
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
Continuity
Differentiability
Proofs in Group Theory
Binary Operations
Groups
Permutation Groups
Fundamental Properties of Groups
Subgroups
Isomorphic Groups
Answers and Hints to Selected Odd-Numbered Exercises
References Index of Symbols
Index of Mathematical Terms

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