# Lecture Slides

SES # COURSEWARE LECTURE SLIDES
Unit 1: Proofs
1 1.1 Intro to Proofs

Welcome to 6.042J (PDF)

Introduction to Proofs (PDF)

2 1.2 Proof Methods

Proof by Cases (PDF)

Proof by Cases Example (PDF)

3 1.3 Well Ordering Principle

Well Ordering Principle 1 (PDF)

Well Ordering Principle 2 (PDF)

Well Ordering Principle 3 (PDF)
4 1.4 Logic & Propositions

Propositional Operators (PDF)

Digital Logic (PDF)

Truth Tables (PDF)

Implies (PDF)

Propositional Logic (PDF)

5 1.5 Quantifiers & Predicate Logic

Predicate Logic 1 (PDF)

Predicate Logic 2 (PDF)

Predicate Logic 3 (PDF)

6 1.6 Sets

Sets Definition (PDF)

Sets Operation (PDF)

7 1.7 Binary Relations

Relations (PDF)

Relational Mapping (PDF)

Finite Cardinality (PDF)

8 1.8 Induction

Induction (PDF)

Bogus Induction (PDF - 1.2MB)

Strong Induction (PDF)

Ordinary Induction vs Strong Induction vs WOP (PDF)

9 1.9 State Machines - Invariants

State Machine Invariants (PDF)

Derived Variables (PDF)

10 1.10 Recursive Definition

Recursive Data (PDF)

Structural Induction (PDF)

Recursive Functions (PDF)

11 1.11 Infinite Sets

Cardinality (PDF)

Countable Sets (PDF)

Cantor's Theorem (PDF)

The Halting Problem (PDF)

Set Theory Axioms (PDF)

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)

13 2.2 Congruences

Congruence (PDF)

Inverses Mod N (PDF)

14 2.3 Euler's Theorem

Modular Exponentiation: Euler's Function (PDF)

The Ring Z_n (PDF)

15 2.4 RSA Encryption

RSA Public Key Encryption (PDF)

Reducing Factoring to SAT (PDF)

16 2.5 Digraphs: Walks & Paths

Digraphs: Walks & Paths (PDF)

Digraphs: Connected Vertices (PDF)

17 2.6 Directed Acyclic Graphs (DAGs) & Scheduling

DAGs (PDF)

Scheduling (PDF)

Time vs Processors (PDF)

18 2.7 Partial Orders and Equivalence

Partial Orders (PDF)

Representing Partial Orders As Subset Relations (PDF)

Equivalence Relations (PDF)

19 2.8 Degrees & Isomorphism

Degrees (PDF)

Isomorphism (PDF)

20 2.9 Coloring & Connectivity

Coloring (PDF)

Connectivity (PDF)

k-Connectivity (PDF)

21 2.10 Trees

Trees (PDF)

Tree Coloring (PDF)

Spanning Trees (PDF)

22 2.11 Stable Matching

Stable Matching (PDF - 2.6MB)

Mating Ritual (PDF)

Optimal Stable Matching (PDF)

Bipartite Matching (PDF)

Hall's Theorem (PDF)

Unit 3: Counting
23 3.1 Sums & Products

Arithmetic Sums (PDF)

Geometric Sums (PDF)

Book Stacking (PDF)

Integral Method (PDF)

Stirling's Formula (PDF)

24 3.2 Asymptotics

Asymptotic Notation (PDF)

Asymptotic Properties (PDF)

Asymptotic Blunders (PDF)

25 3.3 Counting with Bijections

Sum and Product Rules (PDF)

26 3.4 Repetitions & Binomial Theorem

Generalized Counting Rules (PDF)

Two Pair Poker Hands (PDF)

Binomial Theorem (PDF)

Bookkeeper Rule, Multinomial Theorem (PDF)

27 3.5 Pigeonhole Principle, Inclusion-Exclusion

The Pigeonhole Principle (PDF)

Inclusion-Exclusion: 2 Set Proof (PDF)

Unit 4: Probability
28 4.1 Intro to Discrete Probability

Tree Model (PDF)

Simplified Monty Hall Tree (PDF)

Sample Spaces (PDF)

29 4.2 Conditional Probability

Conditional Probability (PDF)

Law Of Total Probability (PDF)

Bayes' Theorem (PDF)

Monty Hall Problem (PDF)

30 4.3 Independence & Causality

Independence (PDF)

Mutual Independence (PDF)

31 4.4 Random Variables, Density Functions

Bigger Number Game (PDF)

Random Variables: Independence (PDF)

Random Variables: Uniform & Binomial (PDF)

32 4.5 Expectation

Expectation (PDF)

Total Expectation (PDF)

Mean Time To Failure (PDF)

Linearity Of Expectation (PDF)

33 4.6 Deviation: Markov & Chebyshev Bounds

Deviation From The Mean (PDF)

Markov Bounds (PDF)

Chebyshev Bounds (PDF)

Variance (PDF)

34 4.7 Sampling & Confidence

Law of Large Numbers (PDF)

Independent Sampling Theorem (PDF)

Birthday Matching (PDF)

Sampling and Confidence (PDF)

35 4.8 Random Walks & Pagerank

Random Walks (PDF)

Stationary Distributions (PDF)

Pagerank (PDF)