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Introduction to Proofs:
An Active Exploration of Mathematical Language
Jennifer Firkins Nordstrom
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Front Matter
Colophon
Preface
1
Logic
1.1
What is a Proof?
1.2
Validity and Soundness
1.2
Exercises
1.3
Logical Connectives
1.3
Exercises
1.4
Logical Arguments with Connectives
1.4
Exercises
2
Mathematical Statements
2.1
Introduction to Sets
2.1
Exercises
2.2
Quantifiers
2.2.1
Quantified Statements
2.2.2
Negating Quantified Statements
2.2.3
Multiple Quantifiers
2.2.3
Exercises
2.3
Conditional Statements
2.3
Exercises
3
Basic Proof Techniques
3.1
Direct Proof and Counterexample
3.1
Exercises
3.2
More Direct Proof: Rational Numbers and Divisibility
3.2.1
Rational Numbers
3.2.2
Divisibility
3.2.3
Exercises
3.3
Proof by Cases
3.3
Exercises
3.4
Proof by Contradiction and Contrapositive
3.4.1
Proof by Contradiction
3.4.2
Proof by Contrapositive
3.4.3
Two Classical Proofs
3.4.3
Exercises
4
Proof by Induction
4.1
Summation Notation
4.1
Exercises
4.2
Mathematical Induction
4.2
Exercises
4.3
More Mathematical Induction
4.3
Exercises
4.4
Strong Induction
4.4
Exercises
5
Proofs in Specific Contexts
5.1
Existence and Uniqueness Proofs
5.1
Exercises
5.2
Set Theory
5.2.1
Set Notation
5.2.2
Operations on Sets
5.2.3
Proofs of Set Properties
5.2.3
Exercises
5.3
Functions
5.3
Exercises
5.4
Equivalence Relations
5.4.1
Introduction to Relations
5.4.2
Equivalence Relations
5.4.2
Exercises
5.5
Ordered Sets
5.5
Exercises
6
Additional Topics
6.1
Binomial Theorem
6.1
Exercises
6.2
Cardinality
6.2
Exercises
Backmatter
Index
A
Notation
Colophon
Colophon
Colophon
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