Advanced Mathematical Proofs

          Module 1: Foundations of Mathematical Reasoning

Section 1: Introduction to Proofs  
- Lesson 1: Understanding What is a Proof  
  Module 1, Section 1, Lesson 1: Understanding What is a Proof  
- Lesson 2: The Importance of Definitions  
  Module 1, Section 1, Lesson 2: The Importance of Definitions

Section 2: Basic Logical Statements  
- Lesson 1: Propositions and Logical Connectives  
  Module 1, Section 2, Lesson 1: Propositions and Logical Connectives  
- Lesson 2: Constructing and Interpreting Truth Tables  
  Module 1, Section 2, Lesson 2: Constructing and Interpreting Truth Tables

Section 3: Elements of Set Theory  
- Lesson 1: Fundamental Set Concepts  
  Module 1, Section 3, Lesson 1: Fundamental Set Concepts  
- Lesson 2: Venn Diagrams and Basic Notation  
  Module 1, Section 3, Lesson 2: Venn Diagrams and Basic Notation

Section 4: Understanding Implications in Proofs  
- Lesson 1: Implication, Converse, and Inverse  
  Module 1, Section 4, Lesson 1: Implication, Converse, and Inverse  
- Lesson 2: Necessary and Sufficient Conditions  
  Module 1, Section 4, Lesson 2: Necessary and Sufficient Conditions

Section 5: Introduction to Basic Proof Techniques  
- Lesson 1: Direct Proofs – The Starting Point  
  Module 1, Section 5, Lesson 1: Direct Proofs – The Starting Point  
- Lesson 2: Using Counterexamples to Disprove Claims  
  Module 1, Section 5, Lesson 2: Using Counterexamples to Disprove Claims


Module 2: Direct and Indirect Proof Techniques

Section 1: Overview of Direct Proofs  
- Lesson 1: Structure of a Direct Proof  
  Module 2, Section 1, Lesson 1: Structure of a Direct Proof  
- Lesson 2: Simple Direct Proof Examples  
  Module 2, Section 1, Lesson 2: Simple Direct Proof Examples

Section 2: Detailed Exploration of Direct Proofs  
- Lesson 1: Step-by-Step Direct Reasoning  
  Module 2, Section 2, Lesson 1: Step-by-Step Direct Reasoning  
- Lesson 2: Translating Problems into Direct Proofs  
  Module 2, Section 2, Lesson 2: Translating Problems into Direct Proofs

Section 3: Introduction to Indirect Proofs  
- Lesson 1: Understanding Proof by Contradiction  
  Module 2, Section 3, Lesson 1: Understanding Proof by Contradiction  
- Lesson 2: Examples Illustrating Indirect Methods  
  Module 2, Section 3, Lesson 2: Examples Illustrating Indirect Methods

Section 4: The Contrapositive Approach  
- Lesson 1: Formulating a Contrapositive Statement  
  Module 2, Section 4, Lesson 1: Formulating a Contrapositive Statement  
- Lesson 2: Applying Contrapositive Reasoning in Proofs  
  Module 2, Section 4, Lesson 2: Applying Contrapositive Reasoning in Proofs

Section 5: Integrating Multiple Proof Strategies  
- Lesson 1: Deciding Between Direct and Indirect Approaches  
  Module 2, Section 5, Lesson 1: Deciding Between Direct and Indirect Approaches  
- Lesson 2: Practice Problems Combining Techniques  
  Module 2, Section 5, Lesson 2: Practice Problems Combining Techniques


Module 3: Introduction to Quantifiers and Predicate Logic

Section 1: Universal Quantifiers Fundamentals  
- Lesson 1: Understanding 'For All' Statements  
  Module 3, Section 1, Lesson 1: Understanding 'For All' Statements  
- Lesson 2: Using Universal Quantifiers in Proofs  
  Module 3, Section 1, Lesson 2: Using Universal Quantifiers in Proofs

Section 2: Existential Quantifiers Explained  
- Lesson 1: Understanding 'There Exists' Statements  
  Module 3, Section 2, Lesson 1: Understanding 'There Exists' Statements  
- Lesson 2: Examples Involving Existential Claims  
  Module 3, Section 2, Lesson 2: Examples Involving Existential Claims

Section 3: Translating Verbal Statements into Logical Expressions  
- Lesson 1: Converting Words into Quantified Statements  
  Module 3, Section 3, Lesson 1: Converting Words into Quantified Statements  
- Lesson 2: Identifying Key Logical Components  
  Module 3, Section 3, Lesson 2: Identifying Key Logical Components

Section 4: Scope and Order of Quantifiers  
- Lesson 1: The Importance of Quantifier Order  
  Module 3, Section 4, Lesson 1: The Importance of Quantifier Order  
- Lesson 2: Analyzing Complex Quantifier Structures  
  Module 3, Section 4, Lesson 2: Analyzing Complex Quantifier Structures

Section 5: Applying Predicate Logic in Proofs  
- Lesson 1: Building Proofs with Predicates  
  Module 3, Section 5, Lesson 1: Building Proofs with Predicates  
- Lesson 2: Practice Translating and Validating Statements  
  Module 3, Section 5, Lesson 2: Practice Translating and Validating Statements


Module 4: Building Complex Proofs

Section 1: Organizing the Structure of a Proof  
- Lesson 1: Recognizing the Parts of a Formal Proof  
  Module 4, Section 1, Lesson 1: Recognizing the Parts of a Formal Proof  
- Lesson 2: Outlining Your Proof Before Writing  
  Module 4, Section 1, Lesson 2: Outlining Your Proof Before Writing

Section 2: Breaking Down Problems into Manageable Steps  
- Lesson 1: Decomposing Complex Problems  
  Module 4, Section 2, Lesson 1: Decomposing Complex Problems  
- Lesson 2: Identifying Key Assumptions  
  Module 4, Section 2, Lesson 2: Identifying Key Assumptions

Section 3: Dealing with Multiple Cases in Proofs  
- Lesson 1: Case Analysis and Organization  
  Module 4, Section 3, Lesson 1: Case Analysis and Organization  
- Lesson 2: Examples of Case-based Proofs  
  Module 4, Section 3, Lesson 2: Examples of Case-based Proofs

Section 4: Introduction to Proof by Induction  
- Lesson 1: The Principle of Mathematical Induction  
  Module 4, Section 4, Lesson 1: The Principle of Mathematical Induction  
- Lesson 2: Applying Induction to Simple Problems  
  Module 4, Section 4, Lesson 2: Applying Induction to Simple Problems

Section 5: Avoiding Common Fallacies  
- Lesson 1: Recognizing Logical Fallacies in Proofs  
  Module 4, Section 5, Lesson 1: Recognizing Logical Fallacies in Proofs  
- Lesson 2: Strategies to Strengthen Your Argument  
  Module 4, Section 5, Lesson 2: Strategies to Strengthen Your Argument


Module 5: Refining Proof Writing Skills

Section 1: Clarity and Precision in Mathematical Language  
- Lesson 1: Writing Clearly and Concisely  
  Module 5, Section 1, Lesson 1: Writing Clearly and Concisely  
- Lesson 2: Avoiding Ambiguity in Statements  
  Module 5, Section 1, Lesson 2: Avoiding Ambiguity in Statements

Section 2: Structuring a Formal Proof Document  
- Lesson 1: Standard Formats and Conventions  
  Module 5, Section 2, Lesson 1: Standard Formats and Conventions  
- Lesson 2: Organizing Your Work for Peer Review  
  Module 5, Section 2, Lesson 2: Organizing Your Work for Peer Review

Section 3: Critiquing and Revising Proofs  
- Lesson 1: Techniques for Self-Evaluation  
  Module 5, Section 3, Lesson 1: Techniques for Self-Evaluation  
- Lesson 2: Constructive Feedback and Revision Strategies  
  Module 5, Section 3, Lesson 2: Constructive Feedback and Revision Strategies

Section 4: Identifying and Overcoming Common Pitfalls  
- Lesson 1: Common Mistakes in Proof Construction  
  Module 5, Section 4, Lesson 1: Common Mistakes in Proof Construction  
- Lesson 2: Methods to Avoid Logical Gaps  
  Module 5, Section 4, Lesson 2: Methods to Avoid Logical Gaps

Section 5: Summarizing and Presenting Proof Strategies  
- Lesson 1: Recapping Core Proof Techniques  
  Module 5, Section 5, Lesson 1: Recapping Core Proof Techniques  
- Lesson 2: Communicating Your Proof Effectively  
  Module 5, Section 5, Lesson 2: Communicating Your Proof Effectively