Set Theory — An Interactive Reference

What is a Set?

A set is an unordered collection of distinct elements. Order doesn't matter; duplicates don't exist.

Core Concept

Definition

A set groups distinct objects (elements) with no inherent ordering. Two sets are equal if and only if they contain exactly the same elements.

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

Notation

Membership

Use ∈ to denote that an element belongs to a set, and ∉ for non-membership.

2 ∈ {1,2,3} ✓
5 ∉ {1,2,3} ✓

Size

Cardinality

The cardinality |A| is the number of elements in a set. Infinite sets (like ℕ) have infinite cardinality. The empty set has cardinality 0.

|{a, b, c}| = 3
|∅| = 0

Containment

Subsets

A is a subset of B (A ⊆ B) if every element of A is also in B. A is a proper subset (A ⊂ B) if A ⊆ B and A ≠ B.

{1,2} ⊆ {1,2,3} ✓
{1,2} ⊂ {1,2,3} ✓

Builder Notation

Set-Builder Form

Describe a set by a rule rather than listing elements. Read the vertical bar as "such that."

{x | x ∈ ℤ, x > 0}
= {1, 2, 3, ...}

Power

Power Set

The power set P(A) is the set of ALL subsets of A, including ∅ and A itself. If |A| = n, then |P(A)| = 2ⁿ.

P({1,2}) =
{∅, {1}, {2}, {1,2}}

Key Properties at a Glance

Extensionality

Sets are defined entirely by their members — two sets with the same elements are identical, regardless of how they were described.

A = B ↔
∀x(x∈A ↔ x∈B)

No Duplicates

An element either belongs to a set or it doesn't. Listing it twice has no effect: {1,1,2} = {1,2}.

{a,a,b} = {a,b}

No Ordering

Sets have no inherent sequence. {a,b} and {b,a} are the same set. (For ordered pairs, use tuples.)

{a,b} = {b,a}

Subset Reflexivity

Every set is a subset of itself. This is always true — there is no element in A that isn't in A.

A ⊆ A

Empty Set

∅ is a subset of every set. Since it has no elements, the condition "every element of ∅ is in B" is vacuously true.

∅ ⊆ A (for all A)

Set Operations

The fundamental ways to combine and transform sets. Each has a precise definition and a visual intuition via Venn diagrams.

A ∪ B

Union

All elements that are in A, or in B, or in both. The combined collection with no duplicates.

{1,2} ∪ {2,3} = {1,2,3}

A ∩ B

Intersection

Only the elements that appear in both A and B. The overlap region.

{1,2,3} ∩ {2,3,4} = {2,3}

A \ B

Difference

Elements in A that are not in B. Also written A − B. Asymmetric — order matters.

{1,2,3} \ {2,3,4} = {1}

Aᶜ

Complement

All elements in the universal set U that are not in A. Requires a defined universe.

U={1..5}, A={1,2}
Aᶜ = {3,4,5}

A △ B

Symmetric Difference

Elements in A or B but not both. Equivalent to (A∪B) \ (A∩B). The XOR of sets.

{1,2,3} △ {2,3,4} = {1,4}

A × B

Cartesian Product

All ordered pairs (a,b) where a∈A and b∈B. Forms the basis of relations and functions.

{1,2}×{a,b} =
{(1,a),(1,b),(2,a),(2,b)}

Formal Definitions

Union

An element is in A ∪ B precisely when it belongs to at least one of the sets.

A ∪ B =
{x | x∈A ∨ x∈B}

Intersection

An element is in A ∩ B precisely when it belongs to both sets simultaneously.

A ∩ B =
{x | x∈A ∧ x∈B}

Difference

An element is in A \ B when it's in A but excluded from B.

A \ B =
{x | x∈A ∧ x∉B}

Sym. Difference

In exactly one of the two sets — not in their intersection.

A △ B =
(A∪B) \ (A∩B)

Interactive Explorer

Edit the sets below and click an operation to see the result visualized.

Set A

Comma-separated values (numbers or letters)

Set B

Comma-separated values (numbers or letters)

Union

A ∪ B

All elements from A and B combined — no duplicates.

Result

Laws & Identities

These algebraic laws hold for all sets A, B, C within a universal set U.

Commutativity

Union is commutative	A ∪ B = B ∪ A
Intersection is commutative	A ∩ B = B ∩ A

Associativity

Union is associative	(A ∪ B) ∪ C = A ∪ (B ∪ C)
Intersection is associative	(A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributivity

∩ distributes over ∪	A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
∪ distributes over ∩	A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

De Morgan's Laws

Complement of union	(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
Complement of intersection	(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ

Identity Laws

Union with empty set	A ∪ ∅ = A
Intersection with universe	A ∩ U = A
Union with universe	A ∪ U = U
Intersection with empty set	A ∩ ∅ = ∅

Complement Laws

Complement union (excluded middle)	A ∪ Aᶜ = U
Complement intersection	A ∩ Aᶜ = ∅
Double complement	(Aᶜ)ᶜ = A
Complement of universe	Uᶜ = ∅
Complement of empty set	∅ᶜ = U

Idempotent & Absorption Laws

Union idempotent	A ∪ A = A
Intersection idempotent	A ∩ A = A
Union absorption	A ∪ (A ∩ B) = A
Intersection absorption	A ∩ (A ∪ B) = A

Special Sets

Canonical sets that appear throughout mathematics and computer science.

∅

Empty Set

Contains no elements. The unique set with cardinality 0. A subset of every set.

∅ = {} | |∅| = 0

U

Universal Set

The set of all objects under consideration in a given context. Defines the domain for complements.

Context-dependent

ℕ

Natural Numbers

The counting numbers. Some definitions include 0; others start at 1. Convention varies by field.

{0, 1, 2, 3, ...} or {1, 2, 3, ...}

ℤ

Integers

All whole numbers: positive, negative, and zero. ℕ ⊂ ℤ.

{..., -2, -1, 0, 1, 2, ...}

ℚ

Rational Numbers

Numbers expressible as a ratio p/q where p,q ∈ ℤ and q ≠ 0. ℤ ⊂ ℚ.

1/2, -3/4, 7, 0.333...

ℝ

Real Numbers

The complete number line, including irrationals like π and √2. ℚ ⊂ ℝ.

π, √2, e, -1.5, 0

ℂ

Complex Numbers

Numbers of the form a + bi where i = √(−1). ℝ ⊂ ℂ.

3 + 2i, -i, 5

P(A)

Power Set

The set of all subsets of A. Always has 2^|A| elements. P(∅) = {∅}.

P({a}) = {∅, {a}}

Number Set Hierarchy

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ
Natural ⊂ Integer ⊂ Rational ⊂ Real ⊂ Complex

Quick Quiz

Test your understanding. 8 questions covering definitions, operations, and laws.

Cheat Sheet

A compact reference for all notation and key identities.

Membership & Relations

x ∈ Ax is an element of A

x ∉ Ax is not an element of A

A ⊆ BA is a subset of B

A ⊂ BA is a proper subset of B

A = BA and B have same elements

|A|Cardinality (size) of A

Operations

A ∪ BUnion: in A or B (or both)

A ∩ BIntersection: in both A and B

A \ BDifference: in A, not in B

AᶜComplement: in U, not in A

A △ BSym. diff: in exactly one

A × BCartesian product of A and B

P(A)Power set: all subsets of A

Key Identities

A ∪ ∅ = AIdentity (union)

A ∩ U = AIdentity (intersection)

A ∪ Aᶜ = UExcluded middle

A ∩ Aᶜ = ∅Non-contradiction

(Aᶜ)ᶜ = ADouble complement

A ∪ A = AIdempotent

De Morgan's Laws

(A∪B)ᶜ= Aᶜ ∩ Bᶜ

(A∩B)ᶜ= Aᶜ ∪ Bᶜ

Special Sets

∅Empty set

ℕNatural numbers

ℤIntegers

ℚRational numbers

ℝReal numbers

ℂComplex numbers