A Deep Introduction to Set Theory

A Deep Introduction to Set Theory

The foundational language of all modern mathematics.

Set theory is the backbone of mathematics. Every branch — algebra, geometry, analysis, topology, probability, algorithms, mathematical physics — rests on the structures introduced here.

This article gives a deeper, more complete introduction to the core objects and operations of set theory: sets, elements, subsets, operations, relations, functions, power sets and cardinality.

1. What Is a Set?

A set is a collection of distinct objects, called elements, grouped together into a single mathematical entity.

We denote sets with braces:

A = {1, 2, 3},    B = {x, y, z}.

Membership is written as:

x ∈ A   (x is an element of A)
x ∉ A   (x is not an element of A)

Important points:

  • Sets care about membership, not order: {1, 2} = {2, 1}.
  • Sets do not contain duplicates: {1, 1, 1, 2} = {1, 2}.
  • Sets can contain abstract objects: numbers, functions, points, vectors, shapes, even other sets.

Mathematically, a set is defined entirely by the question: “Which objects are in it?”

2. Ways to Describe Sets

(a) Roster (list) notation

Explicitly list the elements:

E = {2, 4, 6, 8}.

(b) Set-builder notation

Describe the elements using a rule:

E = {x ∈ ℕ : x is even}

This reads: “E is the set of all natural numbers x such that x is even.”
Set-builder notation is essential for describing infinite or rule-based sets.

3. Subsets and Proper Subsets

A set A is a subset of a set B if every element of A is also in B: 

A ⊆ B.

A proper subset is strictly contained: 

A ⊂ B  means  A ⊆ B and A ≠ B.

Example:

{1, 2} ⊂ {1, 2, 3, 4}.

The empty set

The empty set has no elements:

∅ = {}.

Important facts:

∅ ⊆ A   for every set A.
A ⊆ A   for every set A.

These properties are used constantly in proofs.

4. Universal Set and Complements

If we fix a “universe” U (the set of all objects under discussion), then the complement of A is: 

Aᶜ = U \ A.

Example (universe U = ℝ, the real numbers):

A = {x ∈ ℝ : x > 0}
Aᶜ = {x ∈ ℝ : x ≤ 0}.

The complement is always defined relative to the chosen universe U.

5. Basic Set Operations

Sets can be combined and compared using three fundamental operations.

(a) Union

A ∪ B = {x : x ∈ A or x ∈ B}.

(b) Intersection

A ∩ B = {x : x ∈ A and x ∈ B}.

(c) Difference

A \ B = {x ∈ A : x ∉ B}.

Example:

{1, 2} ∪ {2, 3} = {1, 2, 3}
{1, 2} ∩ {2, 3} = {2}
{1, 2, 3} \ {2} = {1, 3}

These operations satisfy many laws analogous to algebra (commutativity, associativity, distributivity) and also give us De Morgan’s laws, for example: 

(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ.

6. Fundamental Number Sets

Some standard sets appear everywhere:

  • ℕ : natural numbers (1, 2, 3, … or sometimes 0, 1, 2, …)
  • ℤ : integers (… −2, −1, 0, 1, 2, …)
  • ℚ : rational numbers (fractions p/q with p, q ∈ ℤ, q ≠ 0)
  • ℝ : real numbers
  • ℂ : complex numbers (a + bi, where a, b ∈ ℝ)

Each of these is a set and can be studied using the language of set theory.

7. Ordered Pairs and Cartesian Products

Ordered pairs

An ordered pair (a, b) is defined so that: 

(a, b) = (c, d)  if and only if  a = c and b = d.

Order now matters: (1, 2) ≠ (2, 1).

Cartesian products

Given sets A and B, the Cartesian product is: 

A × B = {(a, b) : a ∈ A, b ∈ B}.

Examples:

ℝ² = ℝ × ℝ
ℝ³ = ℝ × ℝ × ℝ.

Points in the plane and in space are elements of these Cartesian products. This is the set-theoretic foundation of coordinate geometry and vector spaces.

8. Relations

A relation on a set A is any subset of A × A. In other words, it is a rule that tells us which ordered pairs of elements are “related”.

Examples of relations on ℝ:

  • “x is equal to y” : R = {(x, y) : x = y}
  • “x is less than y” : R = {(x, y) : x < y}
  • “x is congruent to y mod n” : R = {(x, y) : x − y is a multiple of n}

Relations can have special properties:

  • Reflexive: every element is related to itself.
  • Symmetric: if a is related to b, then b is related to a.
  • Transitive: if a is related to b, and b is related to c, then a is related to c.

A relation that is reflexive, symmetric and transitive is called an equivalence relation. Equivalence relations partition a set into equivalence classes, a key idea in modular arithmetic, quotient spaces, and many constructions in algebra and geometry.

9. Functions as Special Relations

A function is a special kind of relation from A to B with two restrictions: 

f : A → B
  • Every element of A has exactly one image in B.
  • No element of A has two different images.

We often describe the action using mapping notation: 

f : x ↦ x².

Functions can be classified by how they behave:

  • Injective (one-to-one): different inputs give different outputs.
  • Surjective (onto): every element of B is an output of the function.
  • Bijective: both injective and surjective (gives a perfect pairing between A and B).

These ideas are fundamental for understanding invertibility, isomorphisms, and the structure of mathematical objects.

10. Power Sets

The power set of A, denoted ℘(A), is the set of all subsets of A: 

℘(A) = {S : S ⊆ A}.

Example:

A = {1, 2}
℘(A) = {∅, {1}, {2}, {1, 2}}.

If A has n elements, then ℘(A) has 2ⁿ elements. This fact is central in combinatorics, logic and probability.

11. Infinite Sets and Cardinality

Set theory also studies the size (or cardinality) of sets. Two sets have the same cardinality if there is a bijection between them.

Examples of sets with the same infinite size (countably infinite):

ℕ  ~  ℤ  ~  ℚ.

Cantor showed that the real numbers ℝ have strictly larger cardinality than ℕ: 

|ℝ| > |ℕ|.

This was a revolutionary result: there are different sizes of infinity.

12. Why Set Theory Matters

Set theory underpins almost every area of mathematics:

  • Analysis: limits, continuity, sequences, convergence.
  • Algebra: groups, rings, fields, vector spaces.
  • Geometry & Topology: spaces, open sets, neighbourhoods.
  • Probability: sample spaces, events, sigma-algebras.
  • Computer Science: data structures, formal languages, logic.
  • Physics: state spaces, configuration spaces, function spaces.

Almost every mathematical object can be viewed as a set equipped with extra structure. Set theory provides the language for describing that structure.

13. Summary

A solid foundation in set theory includes:

  • Sets and elements
  • Subsets and proper subsets
  • The empty set
  • Set-builder notation
  • Unions, intersections, differences, complements
  • Cartesian products and ordered pairs
  • Relations and equivalence relations
  • Functions and function notation
  • Power sets
  • Cardinality and infinite sets

This framework is the grammar of mathematical thought. Once mastered, it transforms the way you understand every branch of mathematics.

© mathematics.proofs

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