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