https://github.com/hacker-code-j/modern-mathematics
Materials for graduate-level mathematics, including topics like set theory, real analysis, topology, linear algebra and abstract algebra.
https://github.com/hacker-code-j/modern-mathematics
advanced-calculus real-analysis set-theory topology
Last synced: 4 months ago
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Materials for graduate-level mathematics, including topics like set theory, real analysis, topology, linear algebra and abstract algebra.
- Host: GitHub
- URL: https://github.com/hacker-code-j/modern-mathematics
- Owner: Hacker-Code-J
- License: mit
- Created: 2024-06-25T14:54:42.000Z (over 1 year ago)
- Default Branch: main
- Last Pushed: 2025-02-25T06:47:23.000Z (12 months ago)
- Last Synced: 2025-02-25T07:41:49.490Z (12 months ago)
- Topics: advanced-calculus, real-analysis, set-theory, topology
- Language: TeX
- Homepage:
- Size: 28.7 MB
- Stars: 7
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# 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:
---
##### 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
- Subspace; Span
- Subgroup
- Homomorphism; Monomorphism; Epimorphism
- Isomorphism
- Kernel and Image
grad-math-11.tex/pdf: Abstract Algebra I
- Cyclic Group
-
- TBA
grad-math-12.tex/pdf: Abstract Algebra II (TBA)
- TBA
---
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](https://www.tug.org/texlive/acquire.html).
2. Navigate to the desired directory.
3. Use the command `pdflatex ` 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](LICENSE) file for details.