From d56191fd7aafb20cd6625322156630bfc58f1997 Mon Sep 17 00:00:00 2001
From: Shoko <shoko@noreply.fbrns.co>
Date: Sat, 14 Mar 2026 14:20:16 +0000
Subject: [PATCH] Initial commit: math notes vault

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 README.md                                     | 49 +++++++++++++++
 algorithms/00-index.md                        | 21 +++++++
 cryptography-prep/00-index.md                 | 24 +++++++
 discrete-math/00-index.md                     | 19 ++++++
 discrete-math/01-sets-basics.md               | 60 ++++++++++++++++++
 graph-theory/00-index.md                      | 21 +++++++
 ...IT_6_1200J_Lec01_Predicates_Sets_Proofs.md | 62 +++++++++++++++++++
 lecture-notes/README.md                       | 12 ++++
 linear-algebra/00-index.md                    | 19 ++++++
 logic-proofs/00-index.md                      | 19 ++++++
 logic-proofs/01-what-is-a-proof.md            | 27 ++++++++
 logic-proofs/02-propositions.md               | 42 +++++++++++++
 logic-proofs/03-axioms-and-godel.md           | 48 ++++++++++++++
 number-theory/00-index.md                     | 20 ++++++
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diff --git a/README.md b/README.md
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+# Math Foundation for Cryptography
+
+Study system combining:
+- **MIT 6.1200J** — Mathematics for Computer Science (Spring 2024)
+- **University at Buffalo CSE 191** — Discrete Structures
+- **University of Toronto CSC110/111** — Computational Thinking
+
+## Folder Structure
+
+- **discrete-math/** — Sets, logic, proofs, induction
+- **linear-algebra/** — Vectors, matrices, operations
+- **number-theory/** — Modular arithmetic, primes, GCD, etc.
+- **logic-proofs/** — Formal reasoning, proof techniques
+- **graph-theory/** — Graphs, trees, algorithms
+- **algorithms/** — Complexity, sorting, searching
+- **cryptography-prep/** — Building blocks for crypto (once foundation is solid)
+- **lecture-notes/** — Raw notes from lectures before organizing
+
+## Study Flow
+
+1. Watch lecture → save raw notes to `lecture-notes/`
+2. Extract key concepts → organize into appropriate topic folder
+3. Link cross-references between topics
+4. System reminds you of old concepts periodically
+5. Practice problems generated from your notes
+
+## Topics Covered
+
+### Foundational (Start Here)
+- [ ] Sets and Logic
+- [ ] Proof Techniques
+- [ ] Induction
+- [ ] Basic Algorithms
+
+### Intermediate  
+- [ ] Discrete Math (combinatorics, graphs)
+- [ ] Linear Algebra (matrices, vectors)
+- [ ] Modular Arithmetic
+- [ ] Number Theory Basics
+
+### Advanced (Path to Crypto)
+- [ ] Prime Numbers and Factorization
+- [ ] GCD and Extended Euclidean Algorithm
+- [ ] Fermat's Little Theorem
+- [ ] Chinese Remainder Theorem
+- [ ] Cryptography Foundations
+
+## Last Updated
+Created: March 12, 2026
diff --git a/algorithms/00-index.md b/algorithms/00-index.md
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+# Algorithms
+
+## Topics
+- Algorithm Analysis (Big O)
+- Sorting Algorithms
+- Searching Algorithms
+- Recursion
+- Dynamic Programming
+- Divide and Conquer
+
+## Resources
+- Toronto CSC110/111
+- MIT 6.1200J Computational sections
+
+## Key Concepts
+(Add as you learn)
+
+## Related
+- [[graph-theory/00-index|Graph Theory]]
+- [[logic-proofs/00-index|Logic & Proofs]]
+- [[cryptography-prep/00-index|Cryptography Prep]]
diff --git a/cryptography-prep/00-index.md b/cryptography-prep/00-index.md
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+# Cryptography Foundations
+
+## Topics
+- One-way Functions
+- Hash Functions
+- RSA (number theory foundation)
+- Symmetric Cryptography (math background)
+- Public Key Cryptography (math requirements)
+
+## Resources
+- Build from: Number Theory, Linear Algebra, Algorithms
+- MIT 6.1200J (as foundation)
+- Will add crypto-specific resources later
+
+## Key Concepts
+(Add as you learn)
+
+## Related
+- [[number-theory/00-index|Number Theory]]
+- [[linear-algebra/00-index|Linear Algebra]]
+- [[algorithms/00-index|Algorithms]]
+
+## Status
+⏳ Not ready yet — build foundation first
diff --git a/discrete-math/00-index.md b/discrete-math/00-index.md
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+# Discrete Mathematics
+
+## Topics
+- Sets and Elements
+- Logic and Proofs
+- Induction
+- Combinatorics
+- Cardinality
+
+## Resources
+- MIT 6.1200J Lectures on Discrete Math
+- Buffalo CSE 191
+
+## Key Concepts
+(Add as you learn)
+
+## Related
+- [[linear-algebra/00-index|Linear Algebra]]
+- [[logic-proofs/00-index|Logic & Proofs]]
diff --git a/discrete-math/01-sets-basics.md b/discrete-math/01-sets-basics.md
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+# Sets: Basics and Operations
+
+**Source:** MIT 6.1200J Lecture 01
+
+## What is a Set?
+A **set** is a collection of objects.
+
+### Properties
+- **No duplicates:** {1, 2, 2, 3} = {1, 2, 3}
+- **Order doesn't matter:** {1, 2, 3} = {3, 1, 2}
+- **Can be infinite:** ℕ = {0, 1, 2, ...}
+- **Can contain other sets:** B = {2, {3, 4}, ∅}
+
+## Basic Notation
+
+### Membership
+- **∈** (element of): 6 ∈ A means "6 is in set A"
+- **∉** (not element of): {1, 2} ∉ A
+
+### Subset
+- **⊆** (subset): S ⊆ T means all elements of S are also in T
+- Example: {1, 2} ⊆ {0, 1, 2, 6}
+
+## Common Sets
+- **ℕ** (Natural numbers): {0, 1, 2, ...}
+- **ℤ** (Integers): {..., -2, -1, 0, 1, 2, ...}
+- **∅** or **{}** (Empty set): The set with no elements
+
+## Set-Builder Notation
+Used to define sets with predicates:
+
+```
+{n ∈ ℕ | isPrime(n)} = {2, 3, 5, 7, 11, ...}
+```
+
+Reads as: "The set of all n in ℕ such that isPrime(n) is true"
+
+Can also use colon instead of vertical bar: `{n ∈ ℕ : isPrime(n)}`
+
+## Set Operations
+
+### Intersection (∩)
+A ∩ B = {elements in both A and B}
+
+### Union (∪)
+A ∪ B = {elements in A or B (or both)}
+
+### Difference (\ or −)
+A \ B = {elements in A that are not in B}
+
+## Ordered Tuples
+When **order matters**, use parentheses (not braces):
+
+- **(6, 1, 2, 0) ≠ (2, 1, 6, 0)**
+- Duplicates allowed: (1, 2, 2, 3) is valid
+- Set operations (∩, ∪, etc.) don't apply to tuples
+
+## Related
+- [[../logic-proofs/00-index|Logic & Proofs]]
+- [[00-index|Discrete Math Index]]
diff --git a/graph-theory/00-index.md b/graph-theory/00-index.md
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+# Graph Theory
+
+## Topics
+- Graphs and Nodes
+- Paths and Connectivity
+- Trees
+- Graph Algorithms (DFS, BFS)
+- Shortest Path
+- Spanning Trees
+
+## Resources
+- MIT 6.1200J
+- Buffalo CSE 191
+- Toronto CSC110/111
+
+## Key Concepts
+(Add as you learn)
+
+## Related
+- [[algorithms/00-index|Algorithms]]
+- [[discrete-math/00-index|Discrete Math]]
diff --git a/lecture-notes/MIT_6_1200J_Lec01_Predicates_Sets_Proofs.md b/lecture-notes/MIT_6_1200J_Lec01_Predicates_Sets_Proofs.md
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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
diff --git a/lecture-notes/README.md b/lecture-notes/README.md
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+# Raw Lecture Notes
+
+Store unorganized lecture notes here, then move key concepts to appropriate topic folders.
+
+## Files
+(New notes appear here as you forward them)
+
+## Organization Workflow
+1. Save raw lecture as `lecture-COURSE-DATE.md`
+2. Extract key concepts
+3. Move relevant parts to topic folders
+4. Keep references here for archival
diff --git a/linear-algebra/00-index.md b/linear-algebra/00-index.md
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+# Linear Algebra
+
+## Topics
+- Vectors and Vector Spaces
+- Matrices
+- Linear Transformations
+- Eigenvalues and Eigenvectors
+- Determinants
+
+## Resources
+- MIT 6.1200J (as it relates to CS)
+- Toronto CSC110/111
+
+## Key Concepts
+(Add as you learn)
+
+## Related
+- [[discrete-math/00-index|Discrete Math]]
+- [[cryptography-prep/00-index|Cryptography Prep]]
diff --git a/logic-proofs/00-index.md b/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]]
diff --git a/logic-proofs/01-what-is-a-proof.md b/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]]
diff --git a/logic-proofs/02-propositions.md b/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]]
diff --git a/logic-proofs/03-axioms-and-godel.md b/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?]]
diff --git a/number-theory/00-index.md b/number-theory/00-index.md
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+# Number Theory
+
+## Topics
+- Divisibility and Primes
+- Modular Arithmetic
+- GCD and Extended Euclidean Algorithm
+- Fermat's Little Theorem
+- Chinese Remainder Theorem
+- Factorization
+
+## Resources
+- Buffalo CSE 191
+- MIT 6.1200J Number Theory sections
+
+## Key Concepts
+(Add as you learn)
+
+## Related
+- [[cryptography-prep/00-index|Cryptography Prep]]
+- [[discrete-math/00-index|Discrete Math]]