What is a set in mathematics?

A set is an unordered collection of distinct objects called elements. Sets are denoted by capital letters and their elements are listed inside curly braces, like A = {1, 2, 3}. Two sets are equal if and only if they contain exactly the same elements, regardless of order or repetition in the listing.

What is the difference between a subset and a proper subset?

A subset A of B means every element of A also belongs to B, allowing A to equal B. A proper subset means A is contained in B but is not equal to B -- at least one element of B is missing from A. Every set is a subset of itself, but no set is a proper subset of itself.

What are union, intersection, and complement?

Union (A union B) collects all elements in either set. Intersection (A intersect B) collects only elements in both sets. Complement (A complement) collects everything in the universal set that is not in A. These three operations are the core tools for combining and separating sets.

What is the difference between finite, countable, and uncountable sets?

A finite set has a bounded number of elements. A countably infinite set has elements that can be listed in a sequence matching the natural numbers, like the integers or rationals. An uncountable set is too large to list -- no sequence can cover all its elements. The real numbers are the standard example of an uncountable set.

The power set of A is the set of all possible subsets of A, including the empty set and A itself. If A has n elements, its power set has 2 to the n elements. For example, the power set of {a, b} is {empty set, {a}, {b}, {a, b}}, which has 4 elements.

Set Theory Terms and Definitions

About This Glossary

This glossary organizes 25 set theory terms into five categories that build from basic definitions through operations and size comparisons.

Fundamentals establishes the starting vocabulary with 5 entries: set, element, empty set, universal set, and Venn diagram. These terms define what sets are, what they contain, and how they are visualized.

Subsets covers 4 entries on containment relationships: subset, proper subset, superset, and power set. These terms describe how sets relate through inclusion and how the collection of all subsets forms a new set.

Relationships addresses 5 entries on how sets compare: equal sets, equivalent sets, disjoint sets, overlapping sets, and partitions. Each relationship describes a structural connection between two or more sets.

Operations defines 5 entries on combining and separating sets: union, intersection, complement, set difference, and symmetric difference. These operations produce new sets from existing ones and form the algebraic backbone of set theory.

Cardinality spans 6 entries on measuring set size: cardinality itself, finite sets, infinite sets, countable sets, and uncountable sets. These terms distinguish different magnitudes of sets, from empty through countably and uncountably infinite.

Each definition includes intuitive explanations, key properties, examples, and links to detailed lesson pages. Use the search bar or category filters above to navigate.

CardinalityFundamentalsOperationsRelationshipsSubsets

Cardinality(5)

Fundamentals(5)

Operations(5)

Relationships(5)

Subsets(4)

24 of 24 terms

Cardinality

Cardinality Finite Set Infinite Set Countable Set Uncountable Set

Fundamentals

Set Element Empty Set Universal Set Venn Diagram

Operations

Union Intersection Complement Set Difference Symmetric Difference

Relationships

Equal Sets Equivalent Sets Disjoint Sets Overlapping Sets Partition

Subsets

Subset Proper Subset Superset Power Set

24 terms

Fundamentals

(5 items)

Set

An unordered collection of distinct objects, denoted

A = \{a_1, a_2, \ldots\}

Set Theory — Definition

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intuitionnotationrelated concepts

A set is the most basic structure in mathematics — a container that holds objects without regard to order or repetition. Two sets with the same elements are identical regardless of how their elements are listed.

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Element

An object

x

belonging to a set

A

, written

x \in A

Set Theory — Relationships

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intuitionnotationrelated concepts

An element is any individual object that belongs to a set. The membership relation

\in

is the most primitive notion in set theory — everything else is built from it.

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Empty Set

The unique set containing no elements, denoted

\emptyset = \{\}

Set Theory — Cardinality and Types

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intuitionpropertiesrelated concepts

The empty set is a valid set that simply has nothing in it. It serves as the identity element for union (

A \cup \emptyset = A

) and as an annihilator for intersection (

A \cap \emptyset = \emptyset

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Universal Set

The set

U

containing all elements under consideration in a given context

Set Theory — Cardinality and Types

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intuitionexamplesrelated concepts

The universal set defines the boundaries of discussion. Every set in the problem is a subset of

U

, and the complement of any set is taken relative to

U

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Venn Diagram

A diagram using overlapping closed curves within a rectangle (representing

U

) to visualize set relationships and operations

Venn Diagrams

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intuitionexamplesrelated concepts

Each closed curve represents a set, the rectangle represents the universal set, and spatial overlaps correspond to intersections. Shading regions illustrates the result of operations like union, complement, and set difference.

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Subsets

(4 items)

Subset

A \subseteq B \iff \forall x\,(x \in A \Rightarrow x \in B)

Subsets

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intuitionpropertiesrelated concepts

A set

A

is a subset of

B

when every element of

A

also belongs to

B

. The subset relation is the fundamental way to compare sets and establish containment hierarchies.

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Proper Subset

A \subset B \iff (A \subseteq B) \land (A \neq B)

Subsets — Proper Subsets

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A proper subset is a subset that is strictly smaller than the containing set — at least one element of

B

does not belong to

A

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Superset

B \supseteq A \iff A \subseteq B

Subsets — Superset

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intuitionnotationrelated concepts

The superset relation is the inverse of the subset relation. Saying

B \supseteq A

is the same as saying

A \subseteq B

, viewed from

B

's perspective.

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Power Set

\mathcal{P}(A) = \{S \mid S \subseteq A\}

Subsets — Power Set

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intuitionpropertiesexamplesrelated concepts

The power set of

A

is the set of all possible subsets of

A

, including the empty set and

A

itself. It captures every way to select elements from

A

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Relationships

(5 items)

Equal Sets

A = B \iff \forall x\,(x \in A \iff x \in B)

Set Relationships — Equal Sets

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intuitionpropertiesexamplesrelated concepts

Two sets are equal when they contain exactly the same elements. Order and repetition are irrelevant — only membership matters.

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Equivalent Sets

A \sim B \iff |A| = |B|

Set Relationships — Equivalent Sets

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intuitionexamplesrelated concepts

Two sets are equivalent when a bijection (one-to-one correspondence) exists between them. They have the same cardinality but may contain entirely different elements.

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Disjoint Sets

A \cap B = \emptyset

Set Relationships — Disjoint Sets

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intuitionexamplesrelated concepts

Two sets are disjoint when they share no elements — their intersection is the empty set. In a Venn diagram, disjoint sets appear as non-overlapping circles.

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Overlapping Sets

A \cap B \neq \emptyset

Set Relationships — Overlapping Sets

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intuitionexamplesrelated concepts

Two sets overlap when they share at least one element — their intersection is non-empty. This is the negation of disjointness.

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Partition

A collection

\{A_1, A_2, \ldots, A_n\}

of non-empty subsets of

S

such that

A_i \cap A_j = \emptyset

for

i \neq j

and

\bigcup_{i=1}^{n} A_i = S

Set Relationships — Partitions

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intuitionexamplesrelated concepts

A partition divides a set into non-overlapping pieces that together cover the entire set. Every element of

S

belongs to exactly one piece.

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Operations

(5 items)

Union

A \cup B = \{x \mid x \in A \text{ or } x \in B\}

Operations — Union

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intuitionpropertiesrelated concepts

The union collects every element that appears in at least one of the sets. The "or" is inclusive — elements belonging to both are included once.

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Intersection

A \cap B = \{x \mid x \in A \text{ and } x \in B\}

Operations — Intersection

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intuitionpropertiesrelated concepts

The intersection isolates the elements that belong to both sets simultaneously. If no such elements exist, the intersection is the empty set and the sets are disjoint.

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Complement

A^c = \{x \in U \mid x \notin A\}

Operations — Complement

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intuitionpropertiesnotationrelated concepts

The complement of

A

is everything in the universal set that does not belong to

A

. It depends entirely on the choice of

U

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Set Difference

A \setminus B = \{x \mid x \in A \text{ and } x \notin B\}

Operations — Set Difference

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intuitionpropertiesnotationrelated concepts

The difference

A \setminus B

removes from

A

everything that also belongs to

B

, leaving only the elements exclusive to

A

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Symmetric Difference

A \triangle B = (A \setminus B) \cup (B \setminus A)

Operations — Symmetric Difference

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intuitionpropertiesrelated concepts

The symmetric difference collects elements that belong to exactly one of the two sets — those in

A

but not

B

, together with those in

B

but not

A

. It excludes the intersection.

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Cardinality

(5 items)

Cardinality

|A|

— the number of elements in

A

Cardinality — Definition

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intuitionnotationexamplesrelated concepts

Cardinality measures the size of a set. For finite sets, it is simply a count. For infinite sets, cardinality distinguishes different "sizes" of infinity through bijections.

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Finite Set

A set

A

such that

|A| = n

for some

n \in \mathbb{N}_0

Cardinality — Finite Sets

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intuitionexamplesrelated concepts

A finite set contains a countable, bounded number of elements. Its cardinality is a non-negative integer. The empty set is the smallest finite set with

|A| = 0

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Infinite Set

A set

A

that cannot be put in bijection with any

\{1, 2, \ldots, n\}

Cardinality — Infinite Sets

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intuitionexamplesrelated concepts

An infinite set has no finite bound on the number of its elements. Infinite sets come in different sizes: countably infinite (like

\mathbb{N}

) and uncountably infinite (like

\mathbb{R}

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Countable Set

A set

A

such that

|A| \leq |\mathbb{N}|

— either finite or in bijection with

\mathbb{N}

Cardinality — Countable Sets

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intuitionexamplesrelated concepts

A countable set is one whose elements can be listed in a sequence, even if the sequence is infinite. Every element will eventually appear at some position in the list.

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Uncountable Set

A set

A

such that

|A| > |\mathbb{N}|

— no bijection with

\mathbb{N}

exists

Cardinality — Uncountable Sets

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intuitionexamplesrelated concepts

An uncountable set is "too large" to be listed in a sequence. No matter how you try to match its elements with natural numbers, some elements will always be missed.

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