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Methods of Proof

This course is currently under construction. The target release date for this course is October.




Build fluency with sets and logic, the most fundamental structures and operations in mathematics. Learn what a proof is and master a variety of techniques for proving mathematical statements.

Upon successful completion of this course, students will have mastered the following:

Sets and Relations

Modular Arithmetic



23 topics
1.1. Introduction to Sets
1.1.1. Introduction to Sets
1.1.2. Special Sets
1.1.3. Equivalent Sets
1.1.4. Set-Builder Notation
1.1.5. Cardinality of Finite Sets
1.1.6. Subsets
1.1.7. Power Sets
1.2. Set Operations
1.2.1. The Complement of a Set
1.2.2. The Union of Sets
1.2.3. The Intersection of Sets
1.2.4. The Difference of Sets
1.2.5. De Morgan's Laws for Sets
1.2.6. The Cartesian Product
1.2.7. Elementary Properties of Union and Intersection
1.2.8. Distributive Properties of Set Operations
1.2.9. Disjoint Sets
1.2.10. Partitions of Sets
1.2.11. Indexed Sets
1.2.12. Indicator Functions
1.2.13. The Maximum and Minimum of a Set
1.3. Sets in the Plane
1.3.1. Interior and Boundary Points
1.3.2. Interiors and Boundaries of Sets
1.3.3. Open and Closed Sets
30 topics
2.4. Compound Statements
2.4.1. Mathematical Statements
2.4.2. The "And" and "Or" Connectives
2.4.3. The "Not" Connective
2.4.4. Logical Equivalence
2.4.5. Associative and Commutative Laws
2.4.6. Distributing Conjunctions and Disjunctions
2.4.7. The Absorption Laws
2.4.8. De Morgan's Laws for Logic
2.4.9. Translating Between Logical and Set Operations
2.5. Implications
2.5.1. Conditional Statements
2.5.2. Logical Equivalence with Conditional Statements
2.5.3. Converse, Inverse, and Contrapositive
2.5.4. Biconditional Statements
2.5.5. Tautologies and Contradictions
2.6. Predicates
2.6.1. Truth Sets of Predicates
2.6.2. The "And" and "Or" Connectives With Predicates
2.6.3. The "Not" Connective With Predicates
2.6.4. Simplifying Expressions With Predicates Using De Morgan's Laws
2.6.5. Conditional and Biconditional Statements With Predicates
2.6.6. Necessary and Sufficient Conditions
2.6.7. Grammatical Constructions for Implications and Biconditionals
2.7. Quantifiers
2.7.1. Universal and Existential Quantifiers
2.7.2. Negating Universal and Existential Statements
2.7.3. Formal and Informal Language
2.7.4. Nested Quantifiers
2.7.5. Negating Statements With Nested Quantifiers
2.7.6. Prenex Normal Form
2.8. Logical Inference
2.8.1. Implication Elimination and Denying the Consequent
2.8.2. Disjunctive Syllogism and Transitivity of Implication
2.8.3. Additional Rules of Logical Inference
Functions & Sequences
11 topics
3.9. Functions
3.9.1. Sets and Functions
3.9.2. Surjections
3.9.3. Injections
3.9.4. Bijections
3.9.5. Into Functions
3.9.6. Floor and Ceiling Functions
3.10. Sequences
3.10.1. The Limit of a Null Sequence
3.10.2. Proving a Sequence Converges to Zero
3.10.3. Proving a Sequence Converges to a Finite Limit
3.10.4. Infinite Limits of Sequences
3.10.5. Proving a Sequence Has an Infinite Limit
15 topics
4.11. Modular Congruence
4.11.1. Modular Congruence
4.11.2. The Addition Property of Modular Arithmetic
4.11.3. Residues
4.11.4. The Multiplication Property of Modular Arithmetic
4.11.5. The Division Property of Modular Arithmetic
4.11.6. Solving Linear Congruences
4.11.7. Solving Advanced Linear Congruences
4.12. Equivalence Relations
4.12.1. Relations
4.12.2. Equivalence Relations on Scalars
4.12.3. Residue Classes
4.12.4. Equivalence Classes with Scalars
4.13. Proving Statements Concerning Relations
4.13.1. Proving Relations in Modular Arithmetic
4.13.2. Proving Set Relations
4.13.3. Proving Relations With Matrices
4.13.4. Partial Order
Number Theory
5 topics
5.14. Divisibility of Integers
5.14.1. The Division Algorithm
5.14.2. The Euclidean Algorithm
5.14.3. The Extended Euclidean Algorithm
5.14.4. Properties of Divisibility
5.14.5. Linear Diophantine Equations
6 topics
6.15. Cardinality of Sets
6.15.1. Infinite Sets of the Same Cardinality
6.15.2. Cardinality of Natural Numbers, Integers, and Rationals
6.15.3. Cantor's Diagonal Argument
6.15.4. Cantor-Bernstein-Schröder Theorem
6.15.5. The Cardinality of the Power Set of Natural Numbers
6.15.6. Cantor's Theorem
49 topics
7.16. Direct Proof
7.16.1. Parity
7.16.2. Proving Parity Statements
7.16.3. Proving Two-Variable Parity Statements
7.16.4. Proof by Cases
7.16.5. Proving Divisibility
7.16.6. Proving Modular Congruence
7.16.7. Proving Divisibility Using Modular Congruence
7.16.8. Proving Elementary Results Involving Unions and Intersections
7.16.9. Direct Proofs Involving Set Complements and Differences
7.16.10. Direct Proofs Involving Cartesian Products
7.16.11. Direct Proofs of Inequalities
7.16.12. Proving the Existence of an Element in a Set
7.16.13. Direct Proofs of Prime Properties
7.16.14. Direct Proofs of Real Number Statements
7.17. Proof by Induction
7.17.1. Proving Sums of Series Using Induction
7.17.2. Proving Inequalities Using Induction
7.17.3. Proving Divisibility Using Induction
7.17.4. Proving Matrix Identities Using Induction
7.17.5. Proving Recurrence Relations Using Strong Induction
7.18. Proof by Counterexample
7.18.1. Proof by Counterexample
7.18.2. Counterexamples Involving Parity
7.18.3. Counterexamples Involving Divisibility
7.18.4. Counterexamples of Real Number Statements
7.18.5. Counterexamples Involving Modular Congruence
7.18.6. Counterexamples Involving Inequalities
7.18.7. Counterexamples Involving Set Equalities
7.18.8. Counterexamples Involving Set Operations
7.18.9. Counterexamples Involving Cartesian Products
7.19. Proof by Contrapositive
7.19.1. Introduction to Proof by Contrapositive
7.19.2. Proving Parity by Contrapositive
7.19.3. Proving Parity Statements With Multiple Variables by Contrapositive
7.19.4. Proving Divisibility by Contrapositive
7.19.5. Proving Modular Congruence by Contrapositive
7.19.6. Proving Real Number Statements by Contrapositive
7.19.7. Proving Inequalities by Contrapositive
7.19.8. Proving Set Inequalities by Contrapositive
7.19.9. Proving Statements Involving Set Operations by Contrapositive
7.20. Proof by Contradiction
7.20.1. Proof by Contradiction
7.20.2. Proving Parity by Contradiction
7.20.3. Proving Real Number Statements by Contradiction
7.20.4. Proving Divisibility by Contradiction
7.20.5. Proving Modular Congruence by Contradiction
7.20.6. Proving Set Equalities by Contradiction
7.20.7. Proving Statements Involving Set Operations by Contradiction
7.20.8. Proving Inequalities by Contradiction
7.20.9. Proving the Existence of Infinitely Many Primes by Contradiction
7.20.10. Proving the Division Algorithm
7.21. Trivial and Vacuous Proofs
7.21.1. Trivial Proofs
7.21.2. Vacuous Proofs