Modern-Mathematics

Introduction

This repository contains comprehensive course materials for graduate-level mathematics, including topics like set theory, real analysis, topology, linear algebra and abstract algebra. The content is structured in TeX documents, which include detailed notes, diagrams, and proofs.

Structure

Core Materials (grad-math-full)

This directory is the core of the repository, featuring detailed documents for various topics in graduate-level mathematics:

• grad-math-[n].tex: Main source files for each topic.
• grad-math-[n].pdf: Compiled PDFs from the TeX files.

Topics Covered in the grad-math-full Folder

The grad-math-full directory contains detailed documents for graduate-level mathematics courses, organized by specific topics in each document:

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Set Theory

grad-math-1.tex/pdf: Set Theory I

• Set, Power Set, Cartesian Product
• Union, Intersection, Complement
• Function, Image, Pre-image
• Injection, Surjection, Bijection
• Axiom of Choice

grad-math-2.tex/pdf: Set Theory II

• Relation, Equivalence Relation
• Equivalence Class, Partition

Advanced Calculus and Topology

grad-math-3.tex/pdf: Advanced Calculus I

• Boundedness, Supremum and Infimum
• Least Upper Bound Property (Completeness Axiom)
• Well-Ordering Principle and Mathematical Induction
• Archimedean Property

grad-math-4.tex/pdf: Advanced Calculus II

• Convergence of Sequences
• Inequality Rule for Absolute Values
• Limit Theorem (Algebraic Property of Limit of Sequence)

grad-math-5.tex/pdf: Topology I

• Topology and Topological Space
• Open Set
• Continuous Mapping
• Distance Function and Metric Space
• Convergence of Sequences; Continuity of Functions

grad-math-6.tex/pdf: Advanced Calculus III

• Limit of a Function
• Continuity of a Function
• Monotone Convergent Theorem (MCT)
• Nested Interval Property (NIP)
• Bolzano-Weierstrass Theorem
• Limit Superior and Limit Inferior

Linear Algebra and Abstract Algebra

grad-math-7.tex/pdf: Algebraic Structures

• Group
• Ring
• Field
• Module
• Vector Space
• Algebra

grad-math-8.tex/pdf: Linear Algebra I

• Linear Combination, Spanning Set
• Linearly Independent and Dependent
• (Hamel) Basis
• Partial Order, POSET
• Total Order (Linear Order), TOSET
• Maximal, Minimal, Hasse Diagram
• Chain, Zorn's Lemma
• Hamel Basis Theorem (Existence of Basis)
• Invariance of Basis Cardinality; Dimension of Vector Space

grad-math-9.tex/pdf: Linear Algebra II

• Uniqueness of Representation with respect to a Basis; Coordinate
• Linear Transformation
• Vector Space Isomorphism (Linear Isomorphism)
• Classification of Vector Space (up to Isomorphism)
• Matrix Representation of a Linear Transformation}
• TBA

grad-math-10.tex/pdf: Linear Algebra to Abstract Algebra (TBA)

• TBA

grad-math-11.tex/pdf: Abstract Algebra I (TBA)

• TBA

grad-math-12.tex/pdf: Abstract Algebra II (TBA)

• TBA

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Each .tex file is accompanied by a .pdf version which is the compiled and formatted version of the LaTeX document. These documents provide an in-depth look at the fundamental and advanced topics necessary for graduate-level studies in mathematics.

Usage

To compile the documents, you need a TeX distribution such as TeX Live:

1. Install TeX Live: Installation Guide.
2. Navigate to the desired directory.
3. Use the command pdflatex <file-name.tex> to compile a file to PDF.

Contributing

Contributions to the repository are welcome. To contribute:

1. Fork the repository.
2. Create a new branch for your changes.
3. Make your changes.
4. Submit a pull request.

License

This repository is licensed under the MIT License - see the LICENSE file for details.