Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Real Analysis ↓


Set Theory and Logic


In the field of mathematics, particularly real analysis, two fundamental concepts that are essential to understand are set theory and logic. These mathematical frameworks allow individuals to define and manipulate collections of objects, establish rigorous proofs, and create a language that enables precise and concise communication of mathematical ideas.

Understanding sets

At the most basic level, a set is simply a collection of individual objects, considered as a unit in its own right. These objects can be anything: numbers, symbols, or even other sets. Here's an example of a simple set:

A = {1, 2, 3, 4, 5}

In this case, A is a set that contains the numbers from 1 to 5. Each number is called an element, or member, of the set A.

Basic set notation

Sets can be represented using different notations:

  • Roster or tabular notation: A set is listed explicitly. For example, B = {a, e, i, o, u} represents the set of vowels.
  • Set-builder notation: A set is defined by a property that its members must satisfy. For example, C = {x | x is a positive integer less than 6}.

Visual representations can be used to represent sets. Consider the following:

A

Here, the circle represents the set A. The sets may also be overlapping or disjoint.

Operations on sets

Like numbers, sets can be combined and transformed in various ways:

Union

The union of two sets is the set of elements that are either in one of the sets or in both. If A = {1, 2, 3} and B = {3, 4, 5}, then the union is given by:

A ∪ B = {1, 2, 3, 4, 5}
A B

Intersection

The intersection of two sets is the set of elements that are common to both sets. Using the same sets A and B, the intersection is:

A ∩ B = {3}

Difference

The difference of sets A and B (denoted A - B or A B) is the set of elements that are in A but not in B:

A - B = {1, 2}

Complement

The complement of a set A (denoted A' or Ac) with respect to the universal set U consists of all elements of U that are not in A.

If U = {1, 2, 3, 4, 5, 6} and A = {1, 2, 3} Then A' = {4, 5, 6}

Visual representation of operations

U (universal set) A

Logic in mathematics

Logic is the backbone of mathematical reasoning and proofs. It provides a framework for deducing new truths from established facts.

Statements and truth values

In logic, a statement or proposition is a sentence that can be true or false, but not both. For example:

  • The number 5 is prime. This is a true statement.
  • Every even number is prime. This is a false statement.

Logical connectives

Logical connectives allow us to create new statements from existing statements:

  • Conjunction: P ∧ Q (P and Q) is true only if both P and Q are true.
  • Disjunction: P ∨ Q (P or Q) is true if at least one of P or Q is true.
  • Negation: If P is false then ¬P (not P) is true.
  • Implication: P → Q (If P, then Q) is false only if P is true and Q is false.
  • Biconditional: P ↔ Q (P if and only if Q) is true if both P and Q are either true or false.

Truth tables

A truth table is a mathematical table used to determine whether a compound statement is true or false. Here is an example for conjunction:

PQP ∧ Q
TTT
TFF
FTF
FFF

Quantifiers in logic

Logical quantifiers express the extent to which a predicate is true over a series of elements:

  • Universal quantifier: The symbol ∀ signifies "for all". For example, ∀x (x > 0) means "for all x, x is greater than 0".
  • Existential quantifier: The symbol ∃ indicates "there exists". For example, ∃x (x < 0) means "there exists an x such that x is less than 0".

Logic and set theory in real analysis

Real analysis uses set theory and logic extensively to define and prove concepts related to limits, continuity, and integrals. For example, when talking about limits:

A sequence {a_n} converges to a limit L if for every positive number ε (epsilon), there exists a positive integer N such that for all n > N, the absolute value |a_n - L| < ε.

∀ε > 0, ∃N ∈ ℕ such that ∀n > N, |a_n - L| < ε

Conclusion

Set theory and logic are fundamental tools that pervade real analysis. Whether it is about defining sequences, working with limits or understanding continuity, the principles of set theory and logic provide the necessary framework to formulate and prove mathematical ideas. Understanding these concepts is important for anyone involved in advanced mathematics and forms a solid foundation for further exploration in real analysis.


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Set Theory and Logic

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