# 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