The Architecture of Truth: A Guide to Mathematical Proofs

Writing Proofs

A guide to mathematical proof techniques — direct proof, proof by contrapositive, proof by contradiction, mathematical induction, and the strong form of induction — with fully worked examples. Read more... 2687 words,🕔13 minutes read, May 14, 2022.

The Natural Numbers. The Binomial Theorem.

From von Neumann ordinals and the Peano axioms to the well-ordering principle, Pascal's triangle, and a full proof of the binomial theorem — building the natural numbers and combinatorics from first principles. Read more... 3247 words,🕔16 minutes read, May 14, 2022.

Euler's Totient Function and Modular Exponentiation

Euler's totient function, Euler's theorem, and the method of repeated squares — from prime factorization formulas and group theory to fast modular exponentiation as used in cryptography. Read more... 2172 words,🕔11 minutes read, Jan 14, 2022.

Integers. Well Ordering Principle. Division & Euclidean Algorithm.

Constructing the integers from the natural numbers, the division algorithm, the greatest common divisor, Bezout's identity, the Euclidean algorithm, Euclid's lemma, and the fundamental theorem of arithmetic. Read more... 3532 words,🕔17 minutes read, Jan 14, 2022.

The Foundation of Mathematics: An Introduction to Set Theory

Sets. Algebra of sets. Distributive laws. Double inclusion. De Morgan's laws. Truth tables. Cardinality. Inclusion-exclusion principle. Power set. Read more... 2963 words,🕔14 minutes read, May 14, 2022.

Cartesian Products, Relations, and Partitions

Cartesian Product. Relations. Definition, examples, and properties. Equivalence relations. Partitions. Equivalence Classes and Partitions. Read more... 3327 words,🕔16 minutes read, Jul 12, 2022.

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