Abstract Algebra

Abstract algebra is the branch of mathematics concerned with studying algebraic structures themselves, such as groups, rings, and fields. These structures consist of a set of elements equipped with one or more binary operations that satisfy a specific list of axioms, or rules. Instead of focusing on solving equations with concrete numbers, abstract algebra examines the underlying properties and symmetries inherent in these systems, allowing mathematicians to prove general results that apply to a wide variety of mathematical objects, from the integers to geometric transformations, in a single, unified framework.

1. Preliminaries and Foundational Concepts
   1. Set Theory
      1. Sets and Elements
         1. Definition of a Set
         2. Notation and Representation
         3. Membership and Subset Relations
      2. Subsets and Power Sets
         1. Definition of a Subset
         2. Proper and Improper Subsets
         3. Power Set and its Properties
      3. Partitions of Sets
         1. Definition of a Partition
         2. Properties of Partitions
      4. Set Operations
         1. Union
            1. Definition and Notation
            2. Properties of Union
         2. Intersection
            1. Definition and Notation
            2. Properties of Intersection
         3. Complement
            1. Relative Complement
            2. Absolute Complement
            3. Properties of Complements
         4. Cartesian Product
            1. Definition and Notation
            2. Properties and Examples
   2. Relations and Functions
      1. Relations
         1. Definition of a Relation
         2. Properties of Relations
            1. Reflexivity
            2. Symmetry
            3. Transitivity
            4. Antisymmetry
      2. Equivalence Relations
         1. Definition and Properties
         2. Examples of Equivalence Relations
      3. Equivalence Classes and Quotient Sets
         1. Definition of Equivalence Class
         2. Partitioning a Set via Equivalence Relations
         3. Construction of Quotient Sets
      4. Partial and Total Orderings
         1. Partial Order Relations
            1. Definition and Examples
            2. Hasse Diagrams
         2. Total Order Relations
            1. Definition and Examples
            2. Well-Ordering Principle
      5. Functions
         1. Definition of a Function
         2. Domain, Codomain, and Range
         3. Injective Functions
         4. Surjective Functions
         5. Bijective Functions
         6. Composition of Functions
            1. Associativity of Composition
            2. Properties of Composition
         7. Inverse Functions
            1. Existence and Uniqueness
            2. Properties of Inverses
   3. Properties of Integers
      1. The Division Algorithm
         1. Statement and Proof
      2. Greatest Common Divisor
         1. Definition and Properties
         2. Existence and Uniqueness
      3. The Euclidean Algorithm
         1. Procedure and Examples
         2. Extended Euclidean Algorithm
      4. Prime Numbers and Unique Factorization
         1. Definition of a Prime Number
         2. Fundamental Theorem of Arithmetic
         3. Existence and Uniqueness of Factorization
      5. Modular Arithmetic
         1. Definition of Congruence Modulo n
         2. Properties of Modular Arithmetic
         3. Addition Modulo n
         4. Subtraction Modulo n
         5. Multiplication Modulo n
      6. Congruence Classes
         1. Definition and Notation
         2. Complete Residue Systems
         3. Arithmetic of Congruence Classes