📐 Module 1: Set Theory
Duration: Weeks 1-2 + Week 6
📚 Core Topics
Weeks 1-2: Foundations
- Set Operations: Union, Intersection, Difference, Symmetric Difference, Complement
- Power Sets: Boolean algebra of sets
- Venn Diagrams: Visual representation
- Cartesian Products
- Paradoxes & Axioms: Russell’s paradox, Zermelo-Fraenkel axioms (ZFC), Axiom of choice
Week 6: Cardinality
- Finite vs Infinite Sets: Understanding size
- Countable & Uncountable Sets: Different infinities
- Pairing Functions: Encodings
- Cantor’s Results: Diagonal argument, Cantor’s theorem
- Classical Theorems: Schroeder-Bernstein theorem
- Paradoxes: Hilbert’s hotel
🔑 Key Concepts
| Concept | Definition | Example |
|---|---|---|
| Power Set | 𝒫(A) = set of all subsets of A | If |A| = n, then |𝒫(A)| = 2ⁿ |
| Finite | |A| = n for some n ∈ ℕ | {1, 2, 3} has cardinality 3 |
| Countable | |A| = |ℕ| | Integers ℤ, Rationals ℚ |
| Uncountable | |A| > |ℕ| | Real numbers ℝ |
💡 Applications
Where you’ll use this:
- 🗄️ Database theory and relational algebra
- 🎲 Probability theory foundations
- 📝 Formal language theory
- 🧮 Algorithm analysis and complexity
✅ Learning Outcomes
By the end of this module, you will be able to:
- Prove set identities using element arguments or algebraic laws
- Construct Venn diagrams for complex expressions
- Determine cardinality of finite and infinite sets
- Prove sets are countable or uncountable
- Apply Cantor’s diagonal argument correctly