
Communicating Mathematics 


01  (12) 


01  (2) 

What Others Have Said About Writing 


03  (1) 


04  (1) 


05  (2) 

Writing Mathematical Expressions 


07  (1) 

Common Words and Phrases in Mathematics 


08  (3) 

Some Closing Comments About Writing 


11  (2) 


13  (16) 


13  (2) 


15  (1) 


16  (2) 


18  (3) 

Indexed Collections of Sets 


21  (2) 


23  (1) 

Cartesian Products of Sets 


24  (5) 


24  (5) 


29  (22) 


29  (2) 

The Negation of a Statement 


31  (1) 

The Disjunction and Conjunction of Statements 


32  (1) 


33  (2) 


35  (1) 


36  (2) 

Tautologies and Contradictions 


38  (1) 


39  (2) 

Some Fundamental Properties of Logical Equivalence 


41  (1) 

Characterizations of Statements 


42  (2) 

Quantified Statements and Their Negatiors 


44  (7) 


46  (5) 

Direct Proof and Proof By Contrapositive 


51  (16) 

Trivial and Vacuous Proofs 


51  (2) 


53  (3) 


56  (4) 


60  (3) 


63  (4) 


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) 


74  (3) 

Fundamental Properties of Set Operations 


77  (2) 

Proofs Involving Cartesian Products of Sets 


79  (4) 


80  (3) 


83  (10) 


83  (1) 

Examples of Proof by Contradiction 


84  (1) 

The Three Prisoners Problem 


85  (2) 

Other Examples of Proof by Contradiction 


87  (1) 


87  (1) 

A Review of the Three Proof Techniques 


88  (5) 


90  (3) 


93  (20) 

Conjectures in Mathematics 


93  (3) 


96  (2) 


98  (2) 

A Review of Negations of Quantified Statements 


100  (1) 


101  (2) 


103  (2) 


105  (2) 

A Quiz of ``Prove or Disprove'' Problems 


107  (6) 


108  (5) 


113  (22) 


113  (1) 

Reflexive, Symmetric, and Transitive Relations 


114  (2) 


116  (3) 

Properties of Equivalence Classes 


119  (4) 


123  (4) 


127  (8) 


130  (5) 


135  (18) 

The Definition of Function 


135  (3) 

The Set of All Functions From A to B 


138  (1) 

Onetoone and Onto Functions 


138  (2) 


140  (3) 


143  (3) 


146  (3) 


149  (4) 


150  (3) 


153  (22) 

The WellOrdering 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) 


171  (4) 


175  (22) 

Numerically Equivalent Sets 


176  (1) 


177  (6) 


183  (5) 

Comparing Cardinalities of Sets 


188  (3) 

The SchroderBernstein Theorem 


191  (6) 


194  (3) 


197  (18) 

Divisibility Properties of Integers 


197  (1) 


198  (4) 


202  (2) 


204  (2) 

Relatively Prime Integers 


206  (2) 

The Fundamental Theorem of Arithmetic 


208  (2) 

Concepts Involving Sums of Divisors 


210  (5) 


211  (4) 


215  (28) 


215  (5) 


220  (4) 


224  (6) 

Fundamental Properties of Limits of Functions 


230  (5) 


235  (2) 


237  (6) 


239  (4) 


243  (26) 


243  (4) 


247  (5) 


252  (3) 

Fundamental Properties of Groups 


255  (2) 


257  (3) 


260  (9) 


263  (6) 
Answers and Hints to Selected OddNumbered Exercises 

269  (12) 
References 

281  (2) 
Index of Symbols 

283  (2) 
Index of Mathematical Terms 

285  