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