Applied Theorem Path

An interactive, code-driven curriculum for the mathematical foundations of machine learning. Concepts are introduced with rigorous definitions and proofs, then made concrete through runnable Jupyter notebooks with finite-set simulations, counterexamples, and visualizations tied to ML.

Structure

curriculum/
└── foundations/            # Foundational math for ML theory
    ├── 00-basic-logic-and-proof-techniques.ipynb
    ├── 01-Cantor_Theorem_IEEE_EN.pdf
    └── 01-sets-functions-and-relations.ipynb

Foundations

00 — Basic Logic and Proof Techniques

Companion notebook covering the logical toolkit used throughout math and ML theory:

• Propositions, implications, and truth tables
• Universal/existential quantifiers and their negations
• Proof techniques: direct, contrapositive, contradiction, induction, construction, cases, counterexamples
• Common confusions (contrapositive vs. converse, vacuous truth)
• Quick drills and exercises with runnable solutions (e.g., PAC-style bound templates)

01 — Sets, Functions, and Relations

Interactive companion to the Sets, Functions, and Relations source:

• Sets and operations; power sets and cardinality
• Functions, images/preimages, composition, and inverses
• Equivalence relations and quotient sets
• Cantor's Theorem (diagonalization) and Schröder–Bernstein Theorem with finite-set simulations
• ML connections: equivalence classes of classifiers, VC dimension of threshold/linear classifiers, Cartesian products in feature spaces
• Relations and partial orders
• Exercises with verified solutions

Usage

Open a notebook and run it interactively:

jupyter notebook curriculum/foundations/

or without a browser:

jupyter nbconvert --to notebook --execute curriculum/foundations/00-basic-logic-and-proof-techniques.ipynb

Dependencies: Python 3 with numpy and matplotlib (used in the visualizations and simulations).