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+# MIT 6.1200J Lecture 01: Predicates, Sets, and Proofs
+**Date:** Tuesday, February 6, 2024  
+**Instructors:** Z. Abel, B. Chapman, E. Demaine
+
+## Course Administration
+- Lectures Tu/Th
+- Recitations W/F (attendance counts 10%)
+- Problem Sets due Mondays, released Tuesdays
+- **Collaboration Policy:** Solve in groups, list collaborators, write solutions independently (no looking at others' work or using communal notes while writing)
+- **Late Policy:** n hours late = 100 - n% of points (min 50%)
+
+## Key Concepts
+
+### What is a Proof?
+**Definition:** A mathematical proof is a verification of a proposition by a chain of logical deductions from a base set of axioms.
+
+### Propositions and Predicates
+- **Proposition:** A statement that is either True or False
+  - Example (True): 2 + 3 = 5
+  - Example (False): 2 + 3 = 6
+- **Predicate:** A proposition whose truth depends on variables
+  - Example: ∀n ∈ ℕ. n² + n + 41 is prime
+
+### Sets (Introduction)
+- **Set:** A collection of objects (no duplicates, order doesn't matter)
+- **Notation:**
+  - ∈ (element of): 6 ∈ A
+  - ⊆ (subset): S ⊆ T means all elements of S are in T
+  - Set-builder notation: {n ∈ ℕ | isPrime(n)} = {2, 3, 5, 7, 11, ...}
+- **Operations:**
+  - ∩ (intersection): A ∩ B
+  - ∪ (union): A ∪ B
+  - \ (difference): A \ B
+- **Ordered Tuples:** (a, b) where order matters and duplicates allowed
+
+#### Common Sets:
+- ℕ = {0, 1, 2, ...} (natural numbers)
+- ℤ = {..., -2, -1, 0, 1, 2, ...} (integers)
+- ∅ or {} (empty set)
+
+### Axioms
+**Definition:** An axiom is a proposition assumed to be True. Must be stated upfront.
+
+**Key Point:** Different axiom systems can be equally valid but yield different results.
+
+#### Example Axioms:
+1. If a = b and b = c, then a = c
+2. **Euclidean:** Given line l and point p ∉ l, exactly one line through p parallel to l
+3. **Hyperbolic:** Given line l and point p ∉ l, infinitely many lines through p parallel to l
+4. **Spherical:** Given line l and point p ∉ l, no line through p parallel to l
+
+### Consistency and Completeness
+- **Consistent:** No proposition can be both proved and disproved
+- **Complete:** Every proposition can be either proved or disproved
+
+**Gödel's Incompleteness Theorem (1930s):** No set of axioms is both complete and consistent.
+
+**Corollary:** If axioms are consistent, there exist true statements that cannot be proved.
+
+## Important Note
+- Understanding someone else's proof ≠ being able to piece your own proof together
+- "In your own words" means: show your reasoning process