| |
| |
| |
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 | |