Mathematical Proofs : A Transition to Advanced Mathematics

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.

Features :

• Proof Strategies. Proofs are often preceded by a discussion, encouraging students to think about the proof before they jump right into it.
• Proof Analysis. Proofs are followed by discussions and reflected upon, to point out certain key details.
• Practice in writing mathematics. Chapter 0 emphasizes effective and clear exposition, correct usage of symbols, writing and displaying mathematical expressions, and using key words and phrases.
• Early introduction of Sets. Coverage of Sets in Chapter 1 helps prepare students for coverage of Logic in Chapter 2.
• Early introduction of Logic. Much of the emphasis in Chapter 2 is on statements, implications, and an introduction to qualified statements. This helps present what is needed to get into proofs as quickly as possible.
• Proof by Contradiction. An entire chapter is devoted to proof by contradiction.
• Exercises. A wide variety of exercises are provided in the text. There are exercises that present students with statements, asking them to decide whether they are true or false. There are proposed proofs of statements, asking if the argument is valid. There are also proofs without a statement given, asking students to supply a statement of what has been proved.
• Writing better proofs. A major goal of the text is to help students learn to construct proofs so that they are not only mathematically correct but are clearly written, convincing, readable, notationally consistent, and grammatically correct.
• Web site for Mathematical Proofs. Three additional chapters, Chapters 14, 15, and 16 (dealing proofs in ring theory, linear algebra, and topology), can be found on the Web site: www.aw.com/chartrand.

0. 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.

1. Sets.

Describing a Set.
Special Sets.
Subsets.
Set Operations.
Indexed Collections of Sets.
Partitions of Sets.
Cartesian Products of Sets.

2. 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.

3. Direct Proof and Proof by Contrapositive.

Trivial and Vacuous Proofs.
Direct Proofs.
Proof by Contrapositive.
Proof by Cases.
Proof Evaluations.

4. 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.

5. Proof by Contradiction.

Proof by Contradiction.
Examples of Proof by Contradiction.
The Three Prisoners Problem.
Other Examples of Proof by Contradiction.
The Irrationality of …À2.
A Review of the Three Proof Techniques.

6. 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.

7. Equivalence Relations.

Relations.
Reflexive, Symmetric, and Transitive Relations.
Equivalence Relations.
Properties of Equivalence Classes.
Congruence Modulo n.
The Integers Modulo n.

8. 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.

9. 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.

10. Cardinalities of Sets.

Numerically Equivalent Sets.
Denumerable Sets.
Uncountable Sets.
Comparing Cardinalities of Sets.
The Schroder-Bernstein Theorem.

11. 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.

12. Proofs in Calculus.

Limits of Sequences.
Infinite Series.
Limits of Functions.
Fundamental Properties of Limits of Functions.
Continuity.
Differentiability.

13. 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.