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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 ï¿½2 | |

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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 | |