Set Operations

Introduction

Sets are the fundamental building blocks of mathematics. A set is a well-defined collection of distinct objects, called elements or members. Set operations allow us to combine, compare, and manipulate sets to form new sets. These operations—union, intersection, complement, and difference—provide the algebraic structure underlying probability theory, logic, database queries, and virtually every branch of mathematics.

Understanding set operations is essential for working with probability (where events are sets of outcomes), logic (where propositions correspond to sets), and computer science (where data structures often represent sets). The laws governing set operations mirror the laws of logic and Boolean algebra.

This page develops the fundamental set operations, establishes their properties through algebraic laws, and illustrates their applications through examples and Venn diagrams.

Basic Notation

Sets are typically denoted by capital letters (A,B,C) and elements by lowercase letters (a,b,x). The notation x∈A means x is an element of A, while x∉A means x is not an element of A.

Sets can be specified by listing elements in braces: A={1,2,3,4,5} or by set-builder notation: A={x:x* is a positive integer less than *6}.

The empty set, denoted ∅ or {}, contains no elements. The universal set U contains all elements under consideration in a given context.

Union

The union of sets A and B, written A∪B, is the set of elements that belong to A or B (or both):

A∪B={x:x∈A* or *x∈B}

The word or is inclusive: elements in both sets are included in the union.

Example

If A={1,2,3,4} and B={3,4,5,6}, then:

A∪B={1,2,3,4,5,6}

Note that elements 3 and 4 appear in both A and B but are listed only once in the union, since sets contain distinct elements.

Intersection

The intersection of sets A and B, written A∩B, is the set of elements that belong to both A and B:

A∩B={x:x∈A* and *x∈B}

The intersection contains only elements common to both sets.

Example

With A={1,2,3,4} and B={3,4,5,6}:

A∩B={3,4}

Disjoint Sets

Two sets are disjoint (or mutually exclusive) if they have no elements in common, meaning A∩B=∅. For example, {1,2} and {3,4} are disjoint.

Complement

The complement of set A, written Ac or (A^′), is the set of all elements in the universal set U that are not in A:

Ac={x∈U:x∉A}

The complement depends on the universal set. If U={1,2,3,4,5,6,7,8,9,10} and A={2,4,6,8,10}, then Ac={1,3,5,7,9}.

Properties of Complement

• (Ac)c=A (double complement)

• A∪Ac=U (a set and its complement cover everything)

• A∩Ac=∅ (a set and its complement share nothing)

• ∅c=U and Uc=∅

Set Difference

The difference of A and B, written A−B or A∖B, is the set of elements in A but not in B:

A−B={x:x∈A* and *x∉B}=A∩Bc

Unlike union and intersection, set difference is not commutative: A−B≠B−A in general.

Example

With A={1,2,3,4} and B={3,4,5,6}:

A−B={1,2} and B−A={5,6}

Symmetric Difference

The symmetric difference of A and B, written A*△*B or A⊕B, is the set of elements in exactly one of the two sets:

A*△*B=(A−B)∪(B−A)=(A∪B)−(A∩B)

The symmetric difference is commutative and associative. It corresponds to the exclusive or (XOR) operation in logic.

Algebraic Laws

Set operations satisfy many algebraic laws analogous to those for numbers or logical operations.

Commutative Laws

A∪B=B∪A and A∩B=B∩A

Associative Laws

(A∪B)∪C=A∪(B∪C) and (A∩B)∩C=A∩(B∩C)

Distributive Laws

A∪(B∩C)=(A∪B)∩(A∪C)

A∩(B∪C)=(A∩B)∪(A∩C)

Both distributive laws hold for sets, unlike for ordinary arithmetic where only one direction works.

De Morgans Laws

De Morgans laws describe how complements interact with union and intersection:

(A∪B)c=Ac∩Bc

(A∩B)c=Ac∪Bc

In words: the complement of a union is the intersection of complements, and the complement of an intersection is the union of complements.

Identity Laws

A∪∅=A and A∩U=A

Domination Laws

A∪U=U and A∩∅=∅

Idempotent Laws

A∪A=A and A∩A=A

Absorption Laws

A∪(A∩B)=A and A∩(A∪B)=A

Subsets and Equality

Set A is a subset of B, written A⊆B, if every element of A is also an element of B. Formally:

A⊆B⟺(∀x)*(x∈A⟹x∈B)

Two sets are equal if and only if each is a subset of the other: A=B iff A⊆B and B⊆A.

A is a proper subset of B, written A⊂B, if A⊆B and A≠B.

Connection to Probability

In probability theory, events are sets of outcomes. The probability axioms use set operations:

P*(A∪B)=P(A)+P(B)−P*(A∩B) (inclusion-exclusion)

P(Ac)=1−P(A)

If A∩B=∅, then P*(A∪B)=P(A)+P(B)

Connection to Logic

Set operations correspond directly to logical operations:

Union (∪) corresponds to OR (∨)

Intersection (∩) corresponds to AND (∧)

Complement (∁) corresponds to NOT (¬)

De Morgan's laws for sets are equivalent to De Morgan's laws for logic. This correspondence is the foundation of Boolean algebra.

Summary

The fundamental set operations are:

• Union A∪B: elements in A or B (or both)

• Intersection A∩B: elements in both A and B

• Complement Ac: elements not in A

• Difference A−B: elements in A but not in B

• Symmetric difference A*△*B: elements in exactly one set

These operations satisfy commutative, associative, distributive, and De Morgan's laws, forming an algebraic structure that underlies probability theory, logic, and computer science.