PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Logic and Foundations ↓


Set Theory


Set theory is a fundamental part of mathematics where we study collections of objects, which we call sets. The theory was introduced by Georg Cantor in the late 19th century. The language and tools of set theory are used in almost every branch of mathematics, making it an indispensable area of study.

What is a set?

A set is a well-defined collection of distinct objects. These objects are called elements or members of the set. Sets are usually denoted by capital letters such as A, B, C, etc. If an element x is in the set A, we write x ∈ A. If x is not in A, we write x ∉ A.

Example: A = {1, 2, 3}

In this example, 1, 2, and 3 are elements of the set A. We say 1 ∈ A, 2 ∈ A, and 3 ∈ A. If we consider the element 4, since it is not in the set A, we write 4 ∉ A.

A 1 2 3

How to describe a set

Sets can be described in a variety of ways, but we will focus on two primary methods: the roster method and the set-builder method.

Rooster method

In the roster method, we list all the elements of the set inside curly braces, separated by commas. For example:

B = {apple, banana, cherry}

This group B contains three fruit elements: apple, banana, and cherry.

Set-builder method

In the set-builder method, we describe the properties or attributes that the elements of the set have in common. This is written using a vertical bar or colon. For example:

C = { x | x is a positive even number }
C = { x : x > 0 and x mod 2 = 0 }

Both descriptions specify the set C as the set of all positive even numbers.

Basic set operations

There are several basic operations we can perform on sets. These include union, intersection, difference, and complement.

Union of sets

The union of two sets is the set of elements that are either in one of the sets or in both. If A and B are sets, then their union is represented by A ∪ B. For example,

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

Intersection of sets

The intersection of two sets is the set of elements that are common to both sets. It is represented by A ∩ B for sets A and B. For example,

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

Difference of sets

The difference of two sets A and B, denoted A - B or A B, is the set of elements that are in A but not in B. For example,

A = {1, 2, 3}
B = {3, 4, 5}
a − b = {1, 2}

The set A - B contains those elements of A that are not in B.

Complement of a set

If U is the universal set, which means that it is the set of all possible elements under consideration, and A is a subset of U, then the complement of A, denoted A' or A c, is the set of elements that are in U but not in A. For example,

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

Venn diagrams

Venn diagrams are a visual way to represent sets and their operations. They consist of circles (representing sets) within a rectangle (representing the universal set). The overlapping regions of the circles show intersections, while the regions outside the overlap show differences.

A B

In this Venn diagram, two overlapping circles represent sets A and B. The overlapping part represents the intersection of A ∩ B.

Subsets and supersets

A set A is called a subset of a set B if every element of A is also an element of B. It is denoted by A ⊆ B. If A is a subset of B but is not equal to B, then A is called a proper subset and is denoted by A ⊂ B.

A = {1, 2}
B = {1, 2, 3, 4}
A ⊆ B

This statement tells us that all the elements of set A are contained in set B. Set A is a proper subset of B because it does not contain all the elements of B.

Power set

The power set of any set S is the set of all possible subsets of S, including the empty set and S itself. The power set is denoted by P(S) or 2 S.

S = {a, b}
P(S) = { {}, {a}, {b}, {a, b} }

For a set with n elements, the power set will have 2 n elements.

Cartesian product

The Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.

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

The Cartesian product can be viewed as a grid or table, with the elements of set A on one axis and the elements of set B on the other, with each bracket representing an ordered pair.

Infinite sets and cardinality

Set theory distinguishes between finite and infinite sets. Infinite sets are sets that do not have a finite number of elements. The cardinality of a set is a measure of the "number of elements" in the set. For finite sets, this is a simple calculation, but infinite sets have infinite cardinality.

For example, the set of natural numbers N = {1, 2, 3, ...} has an infinite number of elements. Georg Cantor demonstrated the surprising result that infinite sets can also have different sizes (cardinalities).

Countable vs. uncountable sets

An infinite set is countable if its elements can be put into one-to-one correspondence with natural numbers. For example, the set of even numbers {2, 4, 6, ...} is countable because every number can be paired with a natural number.

If a set has more elements than the set of natural numbers, then the set is uncountable. The most famous example of an uncountable set is the set of real numbers between 0 and 1.

Applications of set theory

Set theory is used extensively in many areas of mathematics, and also in computer science, logic, and philosophy. Here are some notable applications:

  • Defining a function: A function from a set A to a set B can be viewed as a subset of the Cartesian product A × B
  • Probability: The probability of an event can be viewed as a measure of a set of outcomes in a sample space.
  • Database theory: Operations such as selection and projection in relational databases can be explained using set theory.

Set theory also forms the basis of other important mathematical structures such as groups, rings, and fields.

Beyond its foundational role, set theory remains an area of active research, where mathematicians examine large cardinals, determinacy, and force, among other topics.

Overall, set theory provides the foundation for modern mathematics. It allows mathematicians to work rigorously with infinity and is helpful in developing logical reasoning.


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Set Theory

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