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 → Mathematical Logic and Foundations ↓


Set Theory


Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Although set theory is a vast and deep field, the aim of this article is to provide a comprehensive but simple introduction to the basics of set theory, suitable for graduate level study, while still being accessible due to the simple language used.

Basic concepts of set theory

In set theory, a set is a well-defined collection of distinct objects, considered as an object in its own right. These objects are called the elements or members of the set. Sets are usually denoted by capital letters such as A, B or C and elements are usually denoted by lower case letters such as x, y or z.

The mathematical notation to represent a set is to list its elements between curly braces. For example, the set containing the numbers 1, 2, and 3 is written as {1, 2, 3}. If an element x is a member of a set A, we write this as x ∈ A. Conversely, if an element is not a member of the set, we write x ∉ A.

Visualization of sets

One way to represent sets and their relationships is through diagrams, often called Venn diagrams. Below is a basic example of a set A with 1, 2, and 3 elements.

1 2 3 A

Types of sets

There are several important types of sets in set theory:

  • Empty set: Also known as the null set, it has no elements. It is usually represented by ∅ or {}.
  • Singleton set: A set having only one element.
  • Finite set: A set with a countable number of elements.
  • Infinite set: A set with infinitely many elements.
  • Subset: If all the elements of A are also elements of B, then set A is a subset of set B. It is denoted by A ⊆ B.

Operations on sets

Just like numbers, sets can be combined and manipulated using various operations. Here are the most common set operations:

Union

The union of two sets A and B is a set that contains all the elements of A and B. It is denoted by A ∪ B. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then:

A ∪ B = {1, 2, 3, 4, 5}

Intersection

The intersection of two sets A and B is a set that contains only the elements that are common to both A and B. It is denoted by A ∩ B. For example, with A = {1, 2, 3} and B = {3, 4, 5}:

A ∩ B = {3}

Difference

The difference between two sets A and B, denoted as A - B or A B, is the set that contains those elements of A that are not in B. For example, with A = {1, 2, 3} and B = {3, 4, 5}:

A - B = {1, 2}

Complement

The complement of a set A refers to all the elements not in A, relative to a universal set U containing all the objects in question. The complement of A is denoted A' or U - A.

Cartesian product

The Cartesian product of two sets A and B is the set of all possible ordered pairs (a, b) where a is in A and b is in B. It is denoted by A × B. For example, if A = {1, 2} and B = {x, y}, then:

A × B = {(1, x), (1, y), (2, x), (2, y)}

Power set

The power set of a set A is the set of all possible subsets of A, including the empty set and A itself. The power set is denoted by P(A). For example, if A = {1, 2}, then the power set is:

P(A) = {∅, {1}, {2}, {1, 2}}

Visual example of set operations

Using Venn diagrams, we can beautifully visualize the described operations:

Union and intersect examples

A B A ∩ B

In the above diagram, A ∪ B will cover the entire region of both the circles, whereas A ∩ B will cover only the region where the circles overlap each other.

Applications of set theory

Set theory provides a fundamental language for mathematics and has applications in many areas. Here are some areas where set theory is prominently used:

  • Mathematics: Foundations of number theory, algebra and topology.
  • Computer science: Data structures, databases, and programming languages.
  • Logic: The basis of formal reasoning and reasoning.
  • Statistics: Sample space and probability theory.

Conclusion

This article provides a detailed introduction to the basics of set theory. By understanding what sets are and how to work with them, one can gain a deeper understanding of the many applications of set theory in various scientific fields. Through the use of visual tools such as Venn diagrams and examples, understanding these fundamental concepts becomes intuitive and clear.


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