| Unit 1: Proofs |
| 1 | 1.1 Intro to Proofs |
Welcome to 6.042J (PDF) Introduction to Proofs (PDF)
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| 2 | 1.2 Proof Methods |
Proof by Contradiction (PDF) Proof by Cases (PDF) Proof by Cases Example (PDF)
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| 3 | 1.3 Well Ordering Principle |
Well Ordering Principle 1 (PDF) Well Ordering Principle 2 (PDF) Well Ordering Principle 3 (PDF)
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| 4 | 1.4 Logic & Propositions |
Propositional Operators (PDF) Digital Logic (PDF) Truth Tables (PDF) Implies (PDF) Propositional Logic (PDF)
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| 5 | 1.5 Quantifiers & Predicate Logic |
Predicate Logic 1 (PDF) Predicate Logic 2 (PDF) Predicate Logic 3 (PDF)
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| 6 | 1.6 Sets |
Sets Definition (PDF) Sets Operation (PDF)
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| 7 | 1.7 Binary Relations |
Relations (PDF) Relational Mapping (PDF) Finite Cardinality (PDF)
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| 8 | 1.8 Induction |
Induction (PDF) Bogus Induction (PDF - 1.2MB) Strong Induction (PDF) Ordinary Induction vs Strong Induction vs WOP (PDF)
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| 9 | 1.9 State Machines - Invariants |
State Machine Invariants (PDF) Derived Variables (PDF)
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| 10 | 1.10 Recursive Definition |
Recursive Data (PDF) Structural Induction (PDF) Recursive Functions (PDF)
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| 11 | 1.11 Infinite Sets |
Cardinality (PDF) Countable Sets (PDF) Cantor's Theorem (PDF) The Halting Problem (PDF) Russell's Paradox (PDF) Set Theory Axioms (PDF)
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| Unit 2: Structures |
| 12 | 2.1 GCDs |
GCDs and Linear Combinations (PDF) Euclidean Algorithm (PDF) The Pulverizer (PDF) Die Hard Primes (PDF) Prime Factorization (PDF)
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| 13 | 2.2 Congruences |
Congruence (PDF) Inverses Mod N (PDF)
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| 14 | 2.3 Euler's Theorem |
Modular Exponentiation: Euler's Function (PDF) The Ring Z_n (PDF)
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| 15 | 2.4 RSA Encryption |
RSA Public Key Encryption (PDF) Reducing Factoring to SAT (PDF)
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| 16 | 2.5 Digraphs: Walks & Paths |
Digraphs: Walks & Paths (PDF) Digraphs: Connected Vertices (PDF)
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| 17 | 2.6 Directed Acyclic Graphs (DAGs) & Scheduling |
DAGs (PDF) Scheduling (PDF) Time vs Processors (PDF)
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| 18 | 2.7 Partial Orders and Equivalence |
Partial Orders (PDF) Representing Partial Orders As Subset Relations (PDF) Equivalence Relations (PDF)
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| 19 | 2.8 Degrees & Isomorphism |
Degrees (PDF) Isomorphism (PDF)
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| 20 | 2.9 Coloring & Connectivity |
Coloring (PDF) Connectivity (PDF) k-Connectivity (PDF)
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| 21 | 2.10 Trees |
Trees (PDF) Tree Coloring (PDF) Spanning Trees (PDF)
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| 22 | 2.11 Stable Matching |
Stable Matching (PDF - 2.6MB) Mating Ritual (PDF) Optimal Stable Matching (PDF) Bipartite Matching (PDF) Hall's Theorem (PDF)
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| Unit 3: Counting |
| 23 | 3.1 Sums & Products |
Arithmetic Sums (PDF) Geometric Sums (PDF) Book Stacking (PDF) Integral Method (PDF) Stirling's Formula (PDF)
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| 24 | 3.2 Asymptotics |
Asymptotic Notation (PDF) Asymptotic Properties (PDF) Asymptotic Blunders (PDF)
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| 25 | 3.3 Counting with Bijections |
Sum and Product Rules (PDF) Counting with Bijections (PDF)
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| 26 | 3.4 Repetitions & Binomial Theorem |
Generalized Counting Rules (PDF) Two Pair Poker Hands (PDF) Binomial Theorem (PDF) Bookkeeper Rule, Multinomial Theorem (PDF)
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| 27 | 3.5 Pigeonhole Principle, Inclusion-Exclusion |
The Pigeonhole Principle (PDF) Inclusion-Exclusion: 2 Set Proof (PDF)
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| Unit 4: Probability |
| 28 | 4.1 Intro to Discrete Probability |
Tree Model (PDF) Simplified Monty Hall Tree (PDF) Sample Spaces (PDF)
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| 29 | 4.2 Conditional Probability |
Conditional Probability (PDF) Law Of Total Probability (PDF) Bayes' Theorem (PDF) Monty Hall Problem (PDF)
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| 30 | 4.3 Independence & Causality |
Independence (PDF) Mutual Independence (PDF)
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| 31 | 4.4 Random Variables, Density Functions |
Bigger Number Game (PDF) Random Variables: Independence (PDF) Random Variables: Uniform & Binomial (PDF)
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| 32 | 4.5 Expectation |
Expectation (PDF) Expected Number Of Heads (PDF) Total Expectation (PDF) Mean Time To Failure (PDF) Linearity Of Expectation (PDF)
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| 33 | 4.6 Deviation: Markov & Chebyshev Bounds |
Deviation From The Mean (PDF) Markov Bounds (PDF) Chebyshev Bounds (PDF) Variance (PDF)
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| 34 | 4.7 Sampling & Confidence |
Law of Large Numbers (PDF) Independent Sampling Theorem (PDF) Birthday Matching (PDF) Sampling and Confidence (PDF)
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| 35 | 4.8 Random Walks & Pagerank |
Random Walks (PDF) Stationary Distributions (PDF) Pagerank (PDF)
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