Sets and Set Operations

Introduction

Sets are the foundation of modern mathematics. A set is simply a collection of distinct objects, called elements or members. Despite this simple definition, set theory provides the language in which essentially all of mathematics can be expressed—from numbers to functions to probability spaces.

Set theory was developed in the late 19th century by Georg Cantor and has since become indispensable in computer science (databases, type theory), probability (sample spaces and events), and logic. Understanding sets and their operations is a prerequisite for nearly every advanced mathematical topic.

Basic Definitions

Notation and Membership

Sets are typically denoted by capital letters (A,B,C and elements by lowercase letters(a,b,c). If x is an element of set A we write x∈A (read "x is in A" or "x belongs to A"). If x is not in A we write x∉A

Sets can be defined by listing elements (roster notation): A={1,2,3,4,5} or by describing a property (set-builder notation): A={x∈ℤ:1≤x≤5}

Important Sets

Standard number sets have special symbols:

N={1,2,3,…} — natural numbers (some include 0)

Z={…,−2,−1,0,1,2,…} — integers

Q — rational numbers (fractions p/q where p,q∈Z, q≠0)

R — real numbers

C — complex numbers

The empty set ∅ (or {}) contains no elements. It is a subset of every set.

Subsets and Equality

Set A is a subset of B, written A⊆B, if every element of A is also in B:

A⊆B⟺(∀*x)*(x∈A⇒x∈B)

Two sets are equal if and only if each is a subset of the other:

A=B⟺(A⊆B* and *B⊆A)

A proper subset A⊂B (or A⊊B) means A⊆B and A≠B.

Set Operations

Union

The union of A and B contains all elements in either set:

A∪B={x:x∈A* or *x∈B}

Intersection

The intersection of A and B contains elements in both sets:

A∩B={x:x∈A* and *x∈B}

Sets are disjoint if their intersection is empty: A∩B=∅.

Difference

The difference (or relative complement) A∖B contains elements in A but not in B:

A∖B={x:x∈A* and *x∉B}=A∩Bc

Complement

Given a universal set U containing all objects under consideration, the complement of A is:

Ac=A=U∖A={x∈U:x∉A}

Cartesian Product

The Cartesian product A×B is the set of all ordered pairs:

A×B={(a,b):a∈A* and *b∈B}

For example, R×R=R2 is the Cartesian plane.

Power Set

The power set 𝒫*(A) (or 2) is the set of all subsets of A:

𝒫*(A)={S:S⊆A}

If |A|=n, then |𝒫*(A)|=2. For example,
𝒫*({1,2})={∅,{1},{2},{1,2}}.

Laws of Set Algebra

Set operations satisfy fundamental algebraic properties that parallel logical operations.

Commutative Laws

A∪B=B∪A,A∩B=B∩A

Associative Laws

(A∪B)∪C=A∪(B∪C),(A∩B)∩C=A∩(B∩C)

Distributive Laws

A∩(B∪C)=(A∩B)∪(A∩C)

A∪(B∩C)=(A∪B)∩(A∪C)

De Morgan's Laws

(A∪B)c=Ac∩Bc,(A∩B)c=Ac∪Bc

These laws state that the complement of a union is the intersection of complements, and vice versa.

Identity and Domination

A∪∅=A,A∩U=A (identity)

A∪U=U,A∩∅=∅ (domination)

Complement Laws

A∪Ac=U,A∩Ac=∅,(Ac)c=A

Cardinality

The cardinality |A| of a finite set is the number of elements it contains.

The inclusion–exclusion principle gives the cardinality of a union:

|A∪B|=|A|+|B|−|A∩B|

For three sets:

|A∪B∪C|=|A|+|B|+|C|−|A∩B|−|A∩C|−|B∩C|+|A∩B∩C|

Connection to Other Concepts

In probability, the sample space is a set and events are subsets. The probability axioms and rules (including inclusion-exclusion) mirror set operations.

Functions are defined as special relations, which are subsets of Cartesian products. The domain, codomain, range, and graph of a function are all sets.

In topology, open sets and their properties define the structure of a space. In measure theory, sigma-algebras are collections of sets closed under countable operations.

Summary

A set is a collection of distinct elements. Membership (∈ and subset (⊆ are the fundamental relations. Sets are equal if they have exactly the same elements.

The core operations—union (∪, intersection (∩, complement, and difference—satisfy algebraic laws including commutativity, associativity, distributivity, and De Morgan's laws.

Sets provide the language for mathematics, underpinning everything from number systems to probability to analysis. Mastery of set notation and operations is essential for advanced study.