| Unit 1: Proofs |
| 1 | 1.1 Intro to Proofs | Chapter 1.1–1.6 (PDF) |
Welcome to 6.042J (PDF) Introduction to Proofs (PDF)
| Session 1 In-Class Questions (PDF) | Problem Set 1 (PDF) |
| 2 | 1.2 Proof Methods | Chapter 1.7–1.9 (PDF) |
Proof by Contradiction (PDF) Proof by Cases (PDF) Proof by Cases Example (PDF)
| Session 2 In-Class Questions (PDF) |
| 3 | 1.3 Well Ordering Principle | Chapter 2.1–2.3 (PDF) |
Well Ordering Principle 1 (PDF) Well Ordering Principle 2 (PDF) Well Ordering Principle 3 (PDF)
| Session 3 In-Class Questions (PDF) |
| 4 | 1.4 Logic & Propositions | Chapter 3.1–3.5 (PDF) |
Propositional Operators (PDF) Digital Logic (PDF) Truth Tables (PDF) Implies (PDF) Propositional Logic (PDF)
| Session 4 In-Class Questions (PDF) |
| 5 | 1.5 Quantifiers & Predicate Logic | Chapter 3.6 (PDF) |
Predicate Logic 1 (PDF) Predicate Logic 2 (PDF) Predicate Logic 3 (PDF)
| Session 5 In-Class Questions (PDF) | Problem Set 2 (PDF) |
| 6 | 1.6 Sets | Chapter 4.1–4.2 (PDF) |
Sets Definition (PDF) Sets Operation (PDF)
| Session 6 In-Class Questions (PDF) |
| 7 | 1.7 Binary Relations | Chapter 4.3–4.5 (PDF) |
Relations (PDF) Relational Mapping (PDF) Finite Cardinality (PDF)
| Session 7 In-Class Questions (PDF) | Problem Set 3 (PDF) |
| 8 | 1.8 Induction | Chapter 5.1–5.3 (PDF) |
Induction (PDF) Bogus Induction (PDF - 1.2MB) Strong Induction (PDF) Ordinary Induction vs Strong Induction vs WOP (PDF)
| Session 8 In-Class Questions (PDF) |
| 9 | 1.9 State Machines—Invariants | Chapter 5.4 (PDF) |
State Machine Invariants (PDF) Derived Variables (PDF)
| Session 9 In-Class Questions (PDF) | Problem Set 4 (PDF) |
| 10 | 1.10 Recursive Definition | Chapter 6 (PDF) |
Recursive Data (PDF) Structural Induction (PDF) Recursive Functions (PDF)
| Session 10 In-Class Questions (PDF) |
| 11 | 1.11 Infinite Sets | Chapter 7 (PDF) |
Cardinality (PDF) Countable Sets (PDF) Cantor's Theorem (PDF) The Halting Problem (PDF) Russell's Paradox (PDF) Set Theory Axioms (PDF)
| Session 11 In-Class Questions (PDF) |
| Unit 2: Structures |
| 12 | 2.1 GCDs | Chapter 8.1–8.5 (PDF) |
GCDs and Linear Combinations (PDF) Euclidean Algorithm (PDF) The Pulverizer (PDF) Die Hard Primes (PDF) Prime Factorization (PDF)
| Session 12 In-Class Questions (PDF) | Problem Set 5 (PDF) |
| 13 | 2.2 Congruences | Chapter 8.6–8.9 (PDF) |
Congruence (PDF) Inverses Mod N (PDF)
| Session 13 In-Class Questions (PDF) |
| 14 | 2.3 Euler's Theorem | Chapter 8.10 (PDF) |
Modular Exponentiation: Euler's Function (PDF) The Ring Z_n (PDF)
| Session 14 In-Class Questions (PDF) |
| 15 | 2.4 RSA Encryption | Chapter 8.11–8.12 (PDF) |
RSA Public Key Encryption (PDF) Reducing Factoring to SAT (PDF)
| Session 15 In-Class Questions (PDF) | Problem Set 6 (PDF) |
| 16 | 2.5 Digraphs: Walks & Paths | Chapter 9.1–9.4 (PDF) |
Digraphs: Walks & Paths (PDF) Digraphs: Connected Vertices (PDF)
| Session 16 In-Class Questions (PDF) |
| 17 | 2.6 Directed Acyclic Graphs | Chapter 9.5 (PDF) |
DAGs (PDF) Scheduling (PDF) Time vs Processors (PDF)
| Session 17 In-Class Questions (PDF) | Problem Set 7 (PDF) |
| 18 | 2.7 Partial Orders and Equivalence | Chapter 9.5–9.11 (PDF) |
Partial Orders (PDF) Representing Partial Orders As Subset Relations (PDF) Equivalence Relations (PDF)
| Session 18 In-Class Questions (PDF) |
| 19 | 2.8 Degrees & Isomorphism | Chapter 11.1–11.4 (PDF) |
Degrees (PDF) Isomorphism (PDF)
| Session 19 In-Class Questions (PDF) |
| 20 | 2.9 Coloring & Connectivity | Chapter 11.7–11.9 (PDF) |
Coloring (PDF) Connectivity (PDF) k-Connectivity (PDF)
| Session 20 In-Class Questions (PDF) | Problem Set 8 (PDF) |
| 21 | 2.10 Trees | Chapter 11.9–11.10 (PDF) |
Trees (PDF) Tree Coloring (PDF) Spanning Trees (PDF)
| Session 21 In-Class Questions (PDF) |
| 22 | 2.11 Stable Matching | Chapter 11.6 (PDF) |
Stable Matching (PDF - 2.6MB) Mating Ritual (PDF) Optimal Stable Matching (PDF) Bipartite Matching (PDF) Hall's Theorem (PDF)
| Session 22 In-Class Questions (PDF) |
|
Unit 3: Counting |
| 23 | 3.1 Sums & Products | Chapter 13.1–13.5 (PDF) |
Arithmetic Sums (PDF) Geometric Sums (PDF) Book Stacking (PDF) Integral Method (PDF) Stirling's Formula (PDF)
| Session 23 In-Class Questions (PDF) | Problem Set 9 (PDF) |
| 24 | 3.2 Asymptotics | Chapter 13.7 (PDF) |
Asymptotic Notation (PDF) Asymptotic Properties (PDF) Asymptotic Blunders (PDF)
| Session 24 In-Class Questions (PDF) |
| 25 | 3.3 Counting with Bijections | Chapter 14.1–14.2 (PDF) |
Sum and Product Rules (PDF) Counting with Bijections (PDF)
| Session 25 In-Class Questions (PDF) | Problem Set 10 (PDF) |
| 26 | 3.4 Repetitions & Binomial Theorem | Chapter 14.4–14.7 (PDF) |
Generalized Counting Rules (PDF) Two Pair Poker Hands (PDF) Binomial Theorem (PDF) Bookkeeper Rule, Multinomial Theorem (PDF)
| Session 26 In-Class Questions (PDF) |
| 27 | 3.5 Pigeonhole Principle, Inclusion-Exclusion | Chapter 14.8 (PDF) |
The Pigeonhole Principle (PDF) Inclusion-Exclusion (PDF) Inclusion-Exclusion: 2 Set Proof (PDF)
| Session 27 In-Class Questions (PDF) |
|
Unit 4: Probability |
| 28 | 4.1 Intro to Discrete Probability | Chapter 16.1–16.5 (PDF) |
Tree Model (PDF) Simplified Monty Hall Tree (PDF) Sample Spaces (PDF)
| Session 28 In-Class Questions (PDF) | Problem Set 11 (PDF) |
| 29 | 4.2 Conditional Probability | Chapter 17.1–17.5 (PDF) |
Conditional Probability (PDF) Law Of Total Probability (PDF) Bayes' Theorem (PDF) Monty Hall Problem (PDF)
| Session 29 In-Class Questions (PDF) |
| 30 | 4.3 Independence & Causality | Chapter 17.7–17.8 (PDF) |
Independence (PDF) Mutual Independence (PDF)
| Session 30 In-Class Questions (PDF) | Problem Set 12 (PDF) |
| 31 | 4.4 Random Variables, Density Functions | Chapter 18.1–18.3 (PDF) |
Bigger Number Game (PDF) Random Variables: Independence (PDF) Random Variables: Uniform & Binomial (PDF)
| Session 31 In-Class Questions (PDF) |
| 32 | 4.5 Expectation | Chapter 18.4–18.5 (PDF) |
Expectation (PDF) Expected Number Of Heads (PDF) Total Expectation (PDF) Mean Time To Failure (PDF) Linearity Of Expectation (PDF)
| Session 32 In-Class Questions (PDF) |
| 33 | 4.6 Deviation: Markov & Chebyshev Bounds | Chapter 19.1–19.3 (PDF) |
Deviation From The Mean (PDF) Markov Bounds (PDF) Chebyshev Bounds (PDF) Variance (PDF)
| Session 33 In-Class Questions (PDF) | No Assignment |
| 34 | 4.7 Sampling & Confidence | Chapter 19.4–19.5 (PDF) |
Law of Large Numbers (PDF) Independent Sampling Theorem (PDF) Birthday Matching (PDF) Sampling and Confidence (PDF)
| Session 34 In-Class Questions (PDF) |
| 35 | 4.8 Random Walks & Pagerank | Chapter 20.2 (PDF) |
Random Walks (PDF) Stationary Distributions (PDF) Pagerank (PDF)
| Session 35 In-Class Questions (PDF) |
