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logic-proofs/00-index.md

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+# Logic and Proofs
+
+## Topics
+- Propositional Logic
+- Predicate Logic
+- Proof by Contradiction
+- Mathematical Induction
+- Proof Techniques
+
+## Resources
+- MIT 6.1200J Mathematics for Computer Science
+- Buffalo CSE 191
+
+## Key Concepts
+(Add as you learn)
+
+## Related
+- [[discrete-math/00-index|Discrete Math]]
+- [[algorithms/00-index|Algorithms]]

logic-proofs/01-what-is-a-proof.md

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+# What is a Proof?
+
+**Source:** MIT 6.1200J Lecture 01
+
+## Definition
+A **mathematical proof** is a verification of a proposition by a chain of logical deductions from a base set of axioms.
+
+## Key Components
+1. **Proposition** — A statement that is either True or False
+2. **Axioms** — Assumed-to-be-true base statements
+3. **Logical deductions** — Chain of reasoning connecting axioms to conclusion
+
+## Important: Different Proof Contexts
+- Physics: Experiment/observation
+- Statistics: Sampling
+- Law: Judge/jury verdict
+- Business: Authority
+- Mathematics: Logical deduction from axioms
+
+## Why State Axioms?
+- Mathematics requires assumptions (axiom = stated assumption)
+- Different axiom systems lead to different mathematical worlds
+- Example: Euclidean vs. Hyperbolic geometry — both valid with different parallel axioms
+
+## Related
+- [[../discrete-math/00-index|Discrete Math Foundations]]
+- [[02-propositions|Propositions]]

logic-proofs/02-propositions.md

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+# Propositions and Predicates
+
+**Source:** MIT 6.1200J Lecture 01
+
+## Proposition
+**Definition:** A statement that is either True or False.
+
+### Examples
+- **True:** 2 + 3 = 5
+- **False:** 2 + 3 = 6
+
+### Non-examples (not propositions)
+- "Hello" — not a statement
+- "Who are you?" — a question, not a declarative statement
+
+## Predicate
+**Definition:** A proposition whose truth depends on variables.
+
+### Examples
+- P(n) = "n² + n + 41 is prime" where n ∈ ℕ
+- Q(x, y) = "x + y = 5" where x, y ∈ ℝ
+
+### Notation
+- ∀ (for all): ∀n ∈ ℕ. P(n)
+- ∃ (there exists): ∃n ∈ ℕ. P(n)
+
+## Implication (A ⇒ B)
+**Key point:** A ⇒ B is NOT about causation or time ordering.
+
+Truth table:
+| A | B | A ⇒ B |
+|---|---|-------|
+| T | T | T     |
+| T | F | F     |
+| F | T | **T** |
+| F | F | T     |
+
+**Note:** "If and only if" (A ⟺ B) means (A ⇒ B) AND (B ⇒ A)
+
+## Related
+- [[01-what-is-a-proof|What is a Proof?]]
+- [[03-axioms-and-godel|Axioms & Gödel's Theorem]]

logic-proofs/03-axioms-and-godel.md

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+# Axioms and Gödel's Incompleteness Theorem
+
+**Source:** MIT 6.1200J Lecture 01
+
+## Axiom
+**Definition:** A proposition that is assumed to be True.
+
+### Key Insight
+- You MUST make assumptions in mathematics
+- The key is to **state them upfront**
+
+## Examples of Contradictory Axiom Systems
+All three are equally valid — they define different geometric worlds:
+
+1. **Euclidean:** Given line l and point p ∉ l, exactly one line through p parallel to l
+2. **Hyperbolic:** Given line l and point p ∉ l, infinitely many lines through p parallel to l
+3. **Spherical:** Given line l and point p ∉ l, no line through p parallel to l
+
+**Lesson:** Different axioms yield different proofs and theorems. Anyone who accepts your axioms must accept theorems derived from them.
+
+## Consistency and Completeness
+
+### Consistent
+Definition: A set of axioms is consistent if no proposition can be both proved and disproved.
+
+### Complete
+Definition: A set of axioms is complete if every proposition can be either proved or disproved.
+
+**Ideal scenario:** Both consistent AND complete.
+
+## Gödel's Incompleteness Theorem
+
+**Theorem (Kurt Gödel, 1930s):** No set of axioms is both complete and consistent.
+
+### Impact
+- Shocked the mathematical community
+- Logicians Russell and Whitehead spent careers trying to find complete + consistent axioms for arithmetic — impossible!
+
+### Corollary
+If axioms must be consistent (necessary for validity), then there exist **True statements that cannot be proved**.
+
+### Example
+- Goldbach's Conjecture might be true but unprovable
+- (If so, please don't assign it as homework!)
+
+## Related
+- [[02-propositions|Propositions]]
+- [[01-what-is-a-proof|What is a Proof?]]