Mathematical Foundations | Xingyu Yang

Reading guide

This series builds the language used throughout the mathematics notes. Begin with propositions and proof methods, then study sets and countability, functions, sequences, and finite sums. Number systems explain how arithmetic supports algorithms; inequalities and complex numbers extend the tools available for later problems.

Countability is introduced with sets and then applied through mappings and number systems. Sequences distinguish indices from values, while finite sums supply algebraic tools; systematic convergence tests belong to calculus. For formal semantics, natural deduction, and automated reasoning, continue with the discrete mathematics series after learning the practical proof methods here.

Reading path

Propositions and Axiomatic Systems

Domains, predicates, quantifiers, axioms, and inference rules: the practical language needed to read and write mathematical proofs.

Converse, Inverse, and Contrapositive

Logical equivalence, negation, necessary and sufficient conditions, and complete contrapositive proofs with worked exercises.

Direct Proof

Definitions, witnesses, legal algebraic steps, and five exercises for building complete direct proofs.

Proof by Cases

Exhaustive case splits, remainder classes, boundaries, symmetry, and worked exercises that distinguish cases from separate proof goals.

Indirect Proof

Choosing between contrapositive and contradiction, stating the full negation, and identifying the exact conflicting assumptions.

Mathematical Induction

Starting indices, ordinary and strong induction, recurrence dependencies, and checked sum identities with collapsible solutions.

Naive Set Theory

Membership, set constructions, finite and countable sets, elementwise proofs, and introductory axiomatic comparisons.

Functions and Mappings

Domains, images, injectivity, surjectivity, cardinality comparisons, composition, and inverses.

Sequences and Series

Sequence notation, countable indices and images, arithmetic and geometric sequences, and series through partial sums.

Finite Sums and Summation Techniques

Finite index sets, reindexing, telescoping, summation by parts, and multiple sums with explicit bounds.

Number Systems, Algorithms, and Recursion

Countability of number systems, ordered fields and completeness, integer arithmetic, algorithm invariants, and structural recursion.

Inequalities

Valid inequality transformations, equality conditions, triangle and mean inequalities, Cauchy-Schwarz, rearrangement, and worked applications.

Complex Numbers

Complex arithmetic from real pairs, geometric inequalities, arguments, exponentials, roots, and polynomial factorization, with worked exercises.

Sets, Functions, Sequences, and Summation: Study Guide

A four-chapter route through naive set theory, mappings, sequences, and finite summation.