lOMoARcPSD|58847208
, lOMoARcPSD|58847208
Table of Contents
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0. Communicating Mathematics zl
0.1 Learning Mathematics zl
0.2 What Others Have Said About Writing
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0.3 Mathematical Writing zl
0.4 Using Symbols zl
0.5 Writing Mathematical Expressions zl zl
0.6 Common Words and Phrases in Mathematics zl zl zl zl zl
0.7 Some Closing Comments About Writing
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1. Sets
1.1 Describing a Set zl zl
1.2 Subsets
1.3 Set Operations
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1.4 Indexed Collections of Sets zl zl zl
1.5 Partitions of Sets zl zl
1.6 Cartesian Products of Sets Exercises for Chapter 1 zl zl zl zl zl zl zl
2. Logic
2.1 Statements
2.2 Negations
2.3 Disjunctions and Conjunctions zl zl
2.4 Implications
2.5 More on Implications zl zl
2.6 Biconditionals
2.7 Tautologies and Contradictions zl zl
2.8 Logical Equivalence zl
2.9 Some Fundamental Properties of Logical Equivalence
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2.10 Quantified Statements zl
2.11 Characterizations Exercises for Chapter 2 zl zl zl zl
3. Direct Proof and Proof by Contrapositive
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3.1 Trivial and Vacuous Proofs zl zl zl
3.2 Direct Proofs zl
3.3 Proof by Contrapositive zl zl
3.4 Proof by Cases zl zl
3.5 Proof Evaluations zl z
Exercises for Chapter 3
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4. More on Direct Proof and Proof by Contrapositive
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4.1 Proofs Involving Divisibility of Integers
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4.2 Proofs Involving Congruence of Integers
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4.3 Proofs Involving Real Numbers zl zl zl
4.4 Proofs Involving Sets zl zl
4.5 Fundamental Properties of Set Operations zl zl zl zl
4.6 Proofs Involving Cartesian Products of Sets Exercises for Chapter 4
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5. Existence and Proof by Contradiction
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5.1 Counterexamples
5.2 Proof by Contradiction zl zl
iv
5.3 A Review of Three Proof Techniques
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, lOMoARcPSD|58847208
5.4 Existence Proofs zl
5.5 Disproving Existence Statements Exercises for Chapter 5 zl zl zl zl zl zl
6. Mathematical Induction zl
6.1 The Principle of Mathematical Induction
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6.2 A More General Principle of Mathematical Induction
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6.3 The Strong Principle of Mathematical Induction
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6.4 Proof by Minimum Counterexample Exercises for Chapter 6
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7. Reviewing Proof Techniques zl zl
7.1 Reviewing Direct Proof and Proof by Contrapositive zl zl zl zl zl zl
7.2 Reviewing Proof by Contradiction and Existence Proofs zl zl zl zl zl zl
7.3 Reviewing Induction Proofs zl zl
7.4 Reviewing Evaluations of Proposed Proofs Exercises for Chapter 7zl zl zl zl zl zl zl zl
8. Prove or Disprove
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8.1 Conjectures in Mathematics zl zl
8.2 Revisiting Quantified Statements zl zl
8.3 Testing Statements Exercises for Chapter 8
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9. Equivalence Relations zl
9.1 Relations
9.2 Properties of Relations zl zl
9.3 Equivalence Relations zl
9.4 Properties of Equivalence Classes zl zl zl
9.5 Congruence Modulo n zl zl
9.6 The Integers Modulo n Exercises for Chapter 9
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10. Functions
10.1 The Definition of Function
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10.2 One-to-one and Onto Functions zl zl zl
10.3 Bijective Functions zl
10.4 Composition of Functions zl zl
10.5 Inverse Functions zl zl
Exercises for Chapter 10
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11. Cardinalities of Sets zl zl
11.1 Numerically Equivalent Sets zl zl
11.2 Denumerable Sets zl
11.3 Uncountable Sets zl
11.4 Comparing Cardinalities of Sets zl zl zl
11.5 The Schroder-Bernstein Theorem¨ Exercises for Chapter 11
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12. Proofs in Number Theory
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12.1 Divisibility Properties of Integers zl zl zl
12.2 The Division Algorithm
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12.3 Greatest Common Divisors zl zl
v
12.4 The Euclidean Algorithm
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12.5 Relatively Prime Integers zl zl
12.6 The Fundamental Theorem of Arithmetic
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12.7 Concepts Involving Sums of Divisors Exercises for Chapter 12
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, lOMoARcPSD|58847208
13. Proofs in Combinatorics
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13.1 The Multiplication and Addition Principles
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13.2 The Principle of Inclusion-Exclusion
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13.3 The Pigeonhole Principle
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13.4 Permutations and Combinations zl zl
13.5 The Pascal Triangle
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13.6 The Binomial Theorem
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13.7 Permutations and Combinations with Repetition Exercises for Chapter 13 zl zl zl zl zl zl zl zl
14. Proofs in Calculus
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14.1 Limits of Sequences zl zl
14.2 Infinite Series zl
14.3 Limits of Functions zl zl
14.4 Fundamental Properties of Limits of Functions zl zl zl zl zl
14.5 Continuity
14.6 Differentiability Ex zl
ercises for Chapter 14
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15. Proofs in Group Theory
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15.1 Binary Operations zl
15.2 Groups
15.3 Permutation Groups zl
15.4 Fundamental Properties of Groups zl zl zl
15.5 Subgroups
15.6 Isomorphic Groups Exercises for Chapter 15 zl zl zl zl zl
16. Proofs in Ring Theory (Online)
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16.1 Rings
16.2 Elementary Properties of Rings zl zl zl
16.3 Subrings
16.4 Integral Domains 16.5 Fields zl zl zl zl
Exercises for Chapter 16
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17. Proofs in Linear Algebra (Online)
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17.1 Properties of Vectors in 3-Space zl zl zl zl
17.2 Vector Spaces zl
17.3 Matrices
17.4 Some Properties of Vector Spaces
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17.5 Subspaces
17.6 Spans of Vectors zl zl
17.7 Linear Dependence and Independence zl zl zl
17.8 Linear Transformations zl
17.9 Properties of Linear Transformations zl zl zl zl
Exercises for Chapter 17
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vi
18. Proofs with Real and Complex Numbers (Online)
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18.1 The Real Numbers as an Ordered Field
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18.2 The Real Numbers and the Completeness Axiom
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18.3 Open and Closed Sets of Real Numbers
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18.4 Compact Sets of Real Numbers zl zl zl zl
18.5 Complex Numbers zl
18.6 De Moivre’s Theorem and Euler’s Formula Exercises for Chapter 18
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, lOMoARcPSD|58847208
Table of Contents
zl zl
0. Communicating Mathematics zl
0.1 Learning Mathematics zl
0.2 What Others Have Said About Writing
zl zl zl zl zl
0.3 Mathematical Writing zl
0.4 Using Symbols zl
0.5 Writing Mathematical Expressions zl zl
0.6 Common Words and Phrases in Mathematics zl zl zl zl zl
0.7 Some Closing Comments About Writing
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1. Sets
1.1 Describing a Set zl zl
1.2 Subsets
1.3 Set Operations
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1.4 Indexed Collections of Sets zl zl zl
1.5 Partitions of Sets zl zl
1.6 Cartesian Products of Sets Exercises for Chapter 1 zl zl zl zl zl zl zl
2. Logic
2.1 Statements
2.2 Negations
2.3 Disjunctions and Conjunctions zl zl
2.4 Implications
2.5 More on Implications zl zl
2.6 Biconditionals
2.7 Tautologies and Contradictions zl zl
2.8 Logical Equivalence zl
2.9 Some Fundamental Properties of Logical Equivalence
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2.10 Quantified Statements zl
2.11 Characterizations Exercises for Chapter 2 zl zl zl zl
3. Direct Proof and Proof by Contrapositive
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3.1 Trivial and Vacuous Proofs zl zl zl
3.2 Direct Proofs zl
3.3 Proof by Contrapositive zl zl
3.4 Proof by Cases zl zl
3.5 Proof Evaluations zl z
Exercises for Chapter 3
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4. More on Direct Proof and Proof by Contrapositive
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4.1 Proofs Involving Divisibility of Integers
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4.2 Proofs Involving Congruence of Integers
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4.3 Proofs Involving Real Numbers zl zl zl
4.4 Proofs Involving Sets zl zl
4.5 Fundamental Properties of Set Operations zl zl zl zl
4.6 Proofs Involving Cartesian Products of Sets Exercises for Chapter 4
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5. Existence and Proof by Contradiction
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5.1 Counterexamples
5.2 Proof by Contradiction zl zl
iv
5.3 A Review of Three Proof Techniques
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, lOMoARcPSD|58847208
5.4 Existence Proofs zl
5.5 Disproving Existence Statements Exercises for Chapter 5 zl zl zl zl zl zl
6. Mathematical Induction zl
6.1 The Principle of Mathematical Induction
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6.2 A More General Principle of Mathematical Induction
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6.3 The Strong Principle of Mathematical Induction
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6.4 Proof by Minimum Counterexample Exercises for Chapter 6
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7. Reviewing Proof Techniques zl zl
7.1 Reviewing Direct Proof and Proof by Contrapositive zl zl zl zl zl zl
7.2 Reviewing Proof by Contradiction and Existence Proofs zl zl zl zl zl zl
7.3 Reviewing Induction Proofs zl zl
7.4 Reviewing Evaluations of Proposed Proofs Exercises for Chapter 7zl zl zl zl zl zl zl zl
8. Prove or Disprove
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8.1 Conjectures in Mathematics zl zl
8.2 Revisiting Quantified Statements zl zl
8.3 Testing Statements Exercises for Chapter 8
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9. Equivalence Relations zl
9.1 Relations
9.2 Properties of Relations zl zl
9.3 Equivalence Relations zl
9.4 Properties of Equivalence Classes zl zl zl
9.5 Congruence Modulo n zl zl
9.6 The Integers Modulo n Exercises for Chapter 9
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10. Functions
10.1 The Definition of Function
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10.2 One-to-one and Onto Functions zl zl zl
10.3 Bijective Functions zl
10.4 Composition of Functions zl zl
10.5 Inverse Functions zl zl
Exercises for Chapter 10
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11. Cardinalities of Sets zl zl
11.1 Numerically Equivalent Sets zl zl
11.2 Denumerable Sets zl
11.3 Uncountable Sets zl
11.4 Comparing Cardinalities of Sets zl zl zl
11.5 The Schroder-Bernstein Theorem¨ Exercises for Chapter 11
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12. Proofs in Number Theory
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12.1 Divisibility Properties of Integers zl zl zl
12.2 The Division Algorithm
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12.3 Greatest Common Divisors zl zl
v
12.4 The Euclidean Algorithm
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12.5 Relatively Prime Integers zl zl
12.6 The Fundamental Theorem of Arithmetic
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12.7 Concepts Involving Sums of Divisors Exercises for Chapter 12
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, lOMoARcPSD|58847208
13. Proofs in Combinatorics
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13.1 The Multiplication and Addition Principles
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13.2 The Principle of Inclusion-Exclusion
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13.3 The Pigeonhole Principle
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13.4 Permutations and Combinations zl zl
13.5 The Pascal Triangle
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13.6 The Binomial Theorem
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13.7 Permutations and Combinations with Repetition Exercises for Chapter 13 zl zl zl zl zl zl zl zl
14. Proofs in Calculus
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14.1 Limits of Sequences zl zl
14.2 Infinite Series zl
14.3 Limits of Functions zl zl
14.4 Fundamental Properties of Limits of Functions zl zl zl zl zl
14.5 Continuity
14.6 Differentiability Ex zl
ercises for Chapter 14
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15. Proofs in Group Theory
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15.1 Binary Operations zl
15.2 Groups
15.3 Permutation Groups zl
15.4 Fundamental Properties of Groups zl zl zl
15.5 Subgroups
15.6 Isomorphic Groups Exercises for Chapter 15 zl zl zl zl zl
16. Proofs in Ring Theory (Online)
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16.1 Rings
16.2 Elementary Properties of Rings zl zl zl
16.3 Subrings
16.4 Integral Domains 16.5 Fields zl zl zl zl
Exercises for Chapter 16
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17. Proofs in Linear Algebra (Online)
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17.1 Properties of Vectors in 3-Space zl zl zl zl
17.2 Vector Spaces zl
17.3 Matrices
17.4 Some Properties of Vector Spaces
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17.5 Subspaces
17.6 Spans of Vectors zl zl
17.7 Linear Dependence and Independence zl zl zl
17.8 Linear Transformations zl
17.9 Properties of Linear Transformations zl zl zl zl
Exercises for Chapter 17
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vi
18. Proofs with Real and Complex Numbers (Online)
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18.1 The Real Numbers as an Ordered Field
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18.2 The Real Numbers and the Completeness Axiom
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18.3 Open and Closed Sets of Real Numbers
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18.4 Compact Sets of Real Numbers zl zl zl zl
18.5 Complex Numbers zl
18.6 De Moivre’s Theorem and Euler’s Formula Exercises for Chapter 18
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