Unit 1: Proofs

• ContinueIntro to Proofs

1.1 Intro to Proofs

• 1.1.1 Welcome to 6.042: Video
• 1.1.2 Intro To Proofs: Part 1: Video
• 1.1.3 Intro to Proofs: Part 2: Video
• 1.1.4 Definitions To Know For Proofs
• 1.1.5 Modus Ponens

1.2 Proof Methods

• 1.2.1 Proof By Contradiction: Video
• 1.2.2 Proof By Contradiction
• 1.2.3 Proof By Cases: Video
• 1.2.4 When to Prove by Cases
• 1.2.5 Friends and Strangers
• 1.2.6 A Bogus Proof by Cases
• 1.2.7 A Bogus Proof by Contradiction

1.3 Well Ordering Principle

• 1.3.1 Well Ordering Principle 1: Video
• 1.3.2 Domain for Well Ordering Principle
• 1.3.3 Well Ordering Principle 2: Video
• 1.3.4 Well Ordering Proofs and Counterexamples
• 1.3.5 Well Ordering Principle 3: Video
• 1.3.6 WOP Proof for Geometric Sum
• 1.3.7 Well Ordering Principle - Examples
• 1.3.8 A Bogus Well Ordering Principle Proof

1.4 Logic & Propositions

• 1.4.1 Propositional Operators: Video
• 1.4.2 Propositional Operators
• 1.4.3 Digital Logic: Video
• 1.4.4 Truth Tables: Video
• 1.4.5 Equivalence and Truth Table
• 1.4.6 Implies: Video
• 1.4.7 Propositional Logic: Video
• 1.4.8 Soundness and Validity
• 1.4.9 Logical Connectives

1.5 Quantifiers & Predicate Logic

• 1.5.1 Predicate Logic 1: Video
• 1.5.2 Predicate Logic 2: Video
• 1.5.3 Satisfiability
• 1.5.4 Predicate Logic 3: Video
• 1.5.5 Name That Predicate!
• 1.5.6 Quantifiers
• 1.5.7 Propositions with Quantifiers
• 1.5.8 Predicate Logic

1.6 Sets

• 1.6.1 Sets Definitions: Video
• 1.6.2 Sets Operations: Video
• 1.6.3 Difference

1.7 Binary Relations

• 1.7.1 Relations: Video
• 1.7.2 Range of a Relation
• 1.7.3 Relational Mappings: Video
• 1.7.4 Total Injection
• 1.7.5 Finite Cardinality: Video
• 1.7.6 A inj B
• 1.7.7 Total Relations
• 1.7.8 Surjective Relations
• 1.7.9 Inverse Relations
• 1.7.10 In- ,Sur-, and Bijections
• 1.7.11 Mapping Lemma: Sizes of Domains and Codomains

1.8 Induction

• 1.8.1 Induction: Video
• 1.8.2 Bogus Induction: Video
• 1.8.3 Same Colored Horses
• 1.8.4 Strong Induction: Video
• 1.8.5 Unstacking Game Score
• 1.8.6 WOP vs Induction: Video [optional]
• 1.8.7 Strong vs Ordinary Induction vs WOP [optional]
• 1.8.8 Induction by n+3
• 1.8.9 Induction Rules
• 1.8.10 Postage by Induction
• 1.8.11 A Bogus Induction

1.9 State Machines - Invariants

• 1.9.1 State Machines Invariants: Video
• 1.9.2 State Machine Invariants
• 1.9.3 Derived Variables: Video
• 1.9.4 Derived Variables and Termination
• 1.9.5 Integer Multiplication
• 1.9.6 Chocolate Bars

1.10 Recursive Definition

• 1.10.1 Recursive Data: Video
• 1.10.2 Matching Parentheses
• 1.10.3 Functions of F18
• 1.10.4 Structural Induction: Video
• 1.10.5 Structural Induction: Definition
• 1.10.6 Counting Cases
• 1.10.7 Recursive Functions: Video

1.11 Infinite Sets

• 1.11.1 Cardinality: Video
• 1.11.2 Cantor, Schroeder-Bernstein
• 1.11.3 Countable Sets: Video
• 1.11.4 Cantor's Theorem: Video
• 1.11.5 Cantor's Diagonal Argument
• 1.11.6 Countable and Uncountable Sets
• 1.11.7 The Halting Problem: Video [Optional]
• 1.11.8 Halting Problem Basics [Optional]
• 1.11.9 Russell's Paradox: Video
• 1.11.10 Russell's Paradox [and ZFC optional]
• 1.11.11 Set Theory Axioms: Video [Optional]
• 1.11.12 Set Theory Axioms [Optional]

• ContinueIntro to Proofs