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.

Logic
30 topics

1.1. Statements

1.1.1. | Statements and Predicates | |

1.1.2. | The "And" and "Or" Connectives | |

1.1.3. | The "Not" Connective | |

1.1.4. | Logical Equivalence | |

1.1.5. | The Associative and Commutative Laws | |

1.1.6. | The Distributive Laws | |

1.1.7. | The Absorption Laws | |

1.1.8. | De Morgan's Laws |

1.2. Implications and Biconditionals

1.2.1. | Conditional Statements | |

1.2.2. | Logical Equivalence with Conditional Statements | |

1.2.3. | Biconditional Statements | |

1.2.4. | Tautologies and Contradictions | |

1.2.5. | Converse, Inverse, and Contrapositive |

1.3. Predicates

1.3.1. | Truth Sets of Predicates | |

1.3.2. | The "And" and "Or" Connectives With Predicates | |

1.3.3. | The "Not" Connective With Predicates | |

1.3.4. | Simplifying Predicate Expressions Using De Morgan's Laws | |

1.3.5. | Conditional Statements With Predicates | |

1.3.6. | Necessary and Sufficient Conditions | |

1.3.7. | Grammatical Constructions for Conditional Statements | |

1.3.8. | Translating Between Logical and Set Operations |

1.4. Quantifiers

1.4.1. | Universal and Existential Quantifiers | |

1.4.2. | Negating Quantified Statements | |

1.4.3. | Nested Quantifiers | |

1.4.4. | Formal and Informal Language | |

1.4.5. | Negating Statements With Nested Quantifiers | |

1.4.6. | Prenex Normal Form |

1.5. Logical Inference

1.5.1. | Implication Elimination and Denying the Consequent | |

1.5.2. | Disjunctive Syllogism and Transitivity of Implication | |

1.5.3. | Additional Rules of Logical Inference |

2.

Set Theory
25 topics

2.6. Introduction to Set Theory

2.6.1. | Sets | |

2.6.2. | Special Sets | |

2.6.3. | Equivalent Sets | |

2.6.4. | Set-Builder Notation | |

2.6.5. | Cardinality of Finite Sets | |

2.6.6. | The Maximum and Minimum of a Set |

2.7. Subsets

2.7.1. | Subsets | |

2.7.2. | Power Sets | |

2.7.3. | Partitions of Sets | |

2.7.4. | Indicator Functions | |

2.7.5. | Proving Subset Relations |

2.8. Set Operations

2.8.1. | The Union of Sets | |

2.8.2. | The Intersection of Sets | |

2.8.3. | The Difference of Sets | |

2.8.4. | Set Complements | |

2.8.5. | The Cartesian Product | |

2.8.6. | Indexed Sets | |

2.8.7. | Disjoint Sets |

2.9. Properties of Set Operations

2.9.1. | Elementary Properties of Set Operations | |

2.9.2. | Proving Elementary Properties of Set Operations | |

2.9.3. | De Morgan's Laws for Sets | |

2.9.4. | Proving De Morgan's Laws for Sets | |

2.9.5. | Distributive Properties of Set Operations | |

2.9.6. | Proving Distributive Properties of Union and Intersection | |

2.9.7. | Proving Distributive Properties of the Cartesian Product |

3.

Discrete Mathematics & Number Theory
11 topics

3.10. Integer Divisibility

3.10.1. | Parity | |

3.10.2. | Properties of Divisibility | |

3.10.3. | The Division Algorithm | |

3.10.4. | The Euclidian Algorithm | |

3.10.5. | The Extended Euclidean Algorithm | |

3.10.6. | Linear Diophantine Equations |

3.11. Proof by Induction

3.11.1. | Mathematical Induction | |

3.11.2. | Proving Inequalities Using Induction | |

3.11.3. | Proving Divisibility Using Induction | |

3.11.4. | Proving Matrix Identities Using Induction | |

3.11.5. | Strong Induction and Recurrence Relations |

4.

Mathematical Proof
15 topics

4.12. Introduction to Mathematical Proof

4.12.1. | Direct Proof | |

4.12.2. | Proving Parity | |

4.12.3. | Proving Divisibility | |

4.12.4. | Proof by Cases | |

4.12.5. | Disproving Universal Statements | |

4.12.6. | Disproving Implications | |

4.12.7. | Trivial and Vacuous Proofs | |

4.12.8. | Proving Biconditional Statements |

4.13. Proof by Contrapositive and Contradiction

4.13.1. | Proof by Contrapositive | |

4.13.2. | Proving Parity by Contrapositive | |

4.13.3. | Proving Divisibility by Contrapositive | |

4.13.4. | Proof by Contradiction | |

4.13.5. | Proving Irrationality by Contradiction | |

4.13.6. | Proving Properties of Irrationals by Contradiction | |

4.13.7. | Proving Divisibility by Contradiction |

5.

Sequences & Functions
20 topics

5.14. Sequences

5.14.1. | The Limit of a Null Sequence | |

5.14.2. | Proving the Limit of a Null Sequence | |

5.14.3. | Proving the Finite Limit of a Sequence | |

5.14.4. | Infinite Limits of Sequences | |

5.14.5. | Proving a Sequence Has an Infinite Limit |

5.15. Functions

5.15.1. | Sets and Functions | |

5.15.2. | Injections | |

5.15.3. | Proving Injectivity | |

5.15.4. | Surjections | |

5.15.5. | Proving Surjectivity | |

5.15.6. | Into Functions | |

5.15.7. | Bijections | |

5.15.8. | The Floor and Ceiling Functions |

5.16. Cardinality

5.16.1. | Discrete Infinite Sets With Equal Cardinality | |

5.16.2. | Continuous Infinite Sets With Equal Cardinality | |

5.16.3. | Cardinality of the Natural Numbers, Integers, and Rationals | |

5.16.4. | Cantor's Diagonal Argument | |

5.16.5. | The Cantor-Bernstein-Schröder Theorem | |

5.16.6. | The Cardinality of the Power Set of Natural Numbers | |

5.16.7. | Cantor's Theorem |

6.

Congruence of Integers
12 topics

6.17. Modular Congruence

6.17.1. | Introduction to Modular Congruence | |

6.17.2. | The Addition Property of Modular Arithmetic | |

6.17.3. | Residues | |

6.17.4. | The Multiplication Property of Modular Arithmetic | |

6.17.5. | The Division Property of Modular Arithmetic | |

6.17.6. | Proving Properties of Congruence | |

6.17.7. | Proving Divisibility Using Congruence | |

6.17.8. | Proving Congruence by Contrapositive |

6.18. Linear Congruences

6.18.1. | Solving Linear Congruences | |

6.18.2. | Advanced Linear Congruences | |

6.18.3. | The Chinese Remainder Theorem | |

6.18.4. | Proving Properties of Linear Congruences |

7.

Equivalence Relations
14 topics

7.19. Relations

7.19.1. | Introduction to Relations | |

7.19.2. | The Domain and Range of a Relation | |

7.19.3. | Graphical Representations of Relations | |

7.19.4. | Operations on Relations | |

7.19.5. | N-ary Relations | |

7.19.6. | Functional Relations |

7.20. Equivalence Relations

7.20.1. | Reflexive and Irreflexive Relations | |

7.20.2. | Symmetric and Antisymmetric Relations | |

7.20.3. | Transitive Relations | |

7.20.4. | Equivalence Relations | |

7.20.5. | Proving a Relation Is an Equivalence Relation | |

7.20.6. | Residue Classes | |

7.20.7. | Equivalence Classes | |

7.20.8. | The Integers Modulo N |