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.

- Construct sets using set-builder notation and demonstrate fluency with set operations and terminology.
- Identify surjective, injective, and bijective functions.
- Compute the cardinality of a set and determine whether an infinite set is countable.
- Identify equivalence relations and reason about equivalence classes.

- Define modular congruence in terms of equivalence classes.
- Perform arithmetic operations on residue classes.
- Use the extended euclidean algorithm to compute modular inverses and solve linear diophantine equations.

- Translate between verbal and symbolic forms of mathematical statements.
- Determine whether two statements are equivalent by constructing and comparing truth tables.
- Understand that the contrapositive of a statement is logically equivalent to the original statement.
- Apply rules of logical inference including distribution, absorption, and De Morgan’s laws.
- Translate between logical operations and set operations.
- Understand the difference between necessary and sufficient conditions.
- Translate between formal and informal language using quantifiers.

- Construct direct proofs of statements involving numbers, sets, logical operations, relations, and functions.
- Disprove false statements by finding counterexamples.
- Prove statements using induction, including strong induction.
- Leverage indirect proof techniques, including proof by contradiction and proof by contrapositive, to reformulate a proof statement in a way that is easier to prove.

1.

Sets
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 |

2.

Logic
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 |

3.

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 |

4.

Relations
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 |

5.

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.

Cardinality
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 |

7.

Proof
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 |