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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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