Real Analysis

Real Analysis is the branch of mathematics that provides the rigorous theoretical foundation for the concepts of calculus. It formally investigates the properties of the real number system, sequences, and functions, using precise definitions and logical proofs to establish fundamental ideas such as limits, continuity, differentiation, and integration. By employing tools like the epsilon-delta definition of a limit, this field moves beyond the computational aspects of calculus to build a solid, axiomatic framework that explains why its rules and theorems are valid.

1. Preliminaries: Logic and Set Theory
   1. Basic Logic and Proof Techniques
      1. Propositional Logic
         1. Logical Connectives
            1. Conjunction
            2. Disjunction
            3. Negation
            4. Implication
            5. Biconditional
         2. Truth Tables
            1. Construction of Truth Tables
            2. Evaluation of Complex Statements
         3. Logical Equivalence
            1. Definition and Examples
            2. Common Equivalences
         4. Tautologies and Contradictions
            1. Identification and Applications
      2. Predicate Logic
         1. Quantifiers
            1. Universal Quantifier
            2. Existential Quantifier
            3. Multiple Quantifiers
         2. Negation of Quantified Statements
            1. De Morgan's Laws for Quantifiers
         3. Translating Mathematical Statements
      3. Methods of Proof
         1. Direct Proof
            1. Structure and Strategy
            2. Examples and Practice
         2. Proof by Contradiction
            1. Logical Foundation
            2. Common Applications
            3. Strategy and Technique
         3. Proof by Contrapositive
            1. Logical Equivalence to Direct Proof
            2. When to Use Contrapositive
         4. Proof by Induction
            1. Principle of Mathematical Induction
            2. Base Case and Inductive Step
            3. Strong Induction
            4. Well-Ordering Principle
            5. Common Errors in Induction
   2. Elementary Set Theory
      1. Basic Set Concepts
         1. Set Notation and Terminology
         2. Membership Relations
         3. Set Equality
         4. The Empty Set
         5. Universal Set
      2. Set Relationships
         1. Subset Relations
         2. Proper Subsets
         3. Set Inclusion Properties
      3. Set Operations
         1. Union
            1. Definition and Properties
            2. Generalized Union
         2. Intersection
            1. Definition and Properties
            2. Generalized Intersection
         3. Complement
            1. Absolute Complement
            2. Relative Complement
         4. Set Difference
         5. Symmetric Difference
         6. Cartesian Product
            1. Ordered Pairs
            2. Properties of Cartesian Products
            3. n-fold Cartesian Products
      4. Power Sets
         1. Definition and Construction
         2. Properties of Power Sets
         3. Cardinality Relationships
      5. Relations
         1. Definition of Relations
         2. Properties of Relations
            1. Reflexivity
            2. Symmetry
            3. Transitivity
            4. Antisymmetry
         3. Equivalence Relations
            1. Definition and Examples
            2. Equivalence Classes
            3. Partitions
         4. Order Relations
            1. Partial Orders
            2. Total Orders
            3. Well-Orders
      6. Functions
         1. Definition and Notation
         2. Domain, Codomain, and Range
         3. Function Types
            1. Injective Functions
            2. Surjective Functions
            3. Bijective Functions
         4. Function Operations
            1. Composition of Functions
            2. Inverse Functions
            3. Restrictions and Extensions
      7. Cardinality
         1. Finite Sets
            1. Counting Principles
         2. Infinite Sets
            1. Countably Infinite Sets
            2. Uncountably Infinite Sets
         3. Cardinality Comparisons
            1. Cantor-Bernstein Theorem
         4. Specific Examples
            1. Countability of Natural Numbers
            2. Countability of Integers
            3. Countability of Rational Numbers
            4. Uncountability of Real Numbers
         5. Cantor's Diagonalization
            1. Method and Applications
            2. Power Set Theorem