math-notes/lecture-notes/MIT_6_1200J_Lec01_Predicates_Sets_Proofs.md

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