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 …À2 | |

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 | |

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