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.
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 |
1.2.1. | 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 |
1.3.1. | Interior and Boundary Points | |
1.3.2. | Interiors and Boundaries of Sets | |
1.3.3. | Open and Closed Sets |
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 |
2.5.1. | 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 |
2.6.1. | 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 |
2.7.1. | 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 |
2.8.1. | Implication Elimination and Denying the Consequent | |
2.8.2. | Disjunctive Syllogism and Transitivity of Implication | |
2.8.3. | Additional Rules of Logical Inference |
3.9.1. | 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 |
3.10.1. | 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 |
4.11.1. | 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 |
4.12.1. | Relations | |
4.12.2. | Equivalence Relations on Scalars | |
4.12.3. | Residue Classes | |
4.12.4. | Equivalence Classes with Scalars |
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 |
5.14.1. | 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 |
6.15.1. | 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 |
7.16.1. | 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 |
7.17.1. | 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 |
7.18.1. | 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 |
7.19.1. | 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 |
7.20.1. | 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 |
7.21.1. | Trivial Proofs | |
7.21.2. | Vacuous Proofs |