Mathematical Proofs
By
Gary Chartrand
Western Michigan University
Table of Contents
The Art of Learning and Communicating Mathematics
- Writing Mathematics
- Use of Symbols
- Writing Equations and Formulas
- Labeling and Placement of Figures
- The Use of Certain Words
- Use Good English!
- A Few Quotes and Stories
- Giving a Talk About Mathematics
I. Sets
- Describing a Set
- Special Sets
- Subsets
- Set Operations
- Arbitrary Unions and Intersections
- Partitions
- Cartesian Products
II. Logic
- Statements
- Truth Values
- The Negation of a Statement
- The Disjunction of Statements
- The Conjunction of Statements
- The Implication
- Equivalence of Statements
- Some Fundamental Properties Concerning Statements
- Characterizations
III. Direct and Indirect Proofs
- Trivial and Vacuous Proofs
- Direct Proofs
- Indirect Proofs
- Proof by Cases
IV. Proofs Involving Sets
- Fundamental Properties of Set Operations
- Properties of Cartesian Products of Sets
V. Proof by Contradiction
- Proof by Contradiciton
- Examples of Proof by Contradiction
- The Three Prisoners Problem
- Other Examples of Proof by Contradiction
- The Irrationality of the Square Root of 2
VI. To Prove or Disprove
- University Quantifiers
- Existential Quantifiers
- Negations of Statements Involving Quantifiers
- Counterexamples
VII. Practice Exams on Chapters I-VI
VIII. To Prove or Disprove
- Relations
- Properties of Relations Defined on a Set
- Equivalence Relations
- Congruence Modulo n
- The Integers Modulo n
IX. Functions
- Injective, Surjective, and Bijective Functions
- Composition of Functions
- Inverse Functions
- Permutations
X. Cardinalities of Sets
- Equivalent Sets
- Denumerable Sets
- Nondenumerable Sets
- Sets With High Cardinality
XI. Practice Exams on Chapters VIII-X
XII. Mathematical Induction
- Well-Ordered Sets
- The Principle of Mathematical Induction
- Mathematical Induction and Sums of Numbers
- Mathematical Induction and Inequalities
- Other Examples of Mathematical Induction Proofs
- Proof by Minimum Counterexample
- Variations of Induction
- The Strong Form of Induction
XIII. Proofs in Number Theory
- Primes and Composite Numbers
- The Division Algorithm
- Divisibility Properties of Integers
- Greatest Common Divisors
- Relatively Prime Numbers
- The Infinitude of Primes
XIV. Proofs in Calculus
XV. Proofs in Linear Algebra
XVI. Proofs in Group Theory
XVII. Practice Final Exam