Math Reference

Set Theory Terms and Foundations

Learn sets, operations, relations, functions, infinite cardinalities, axiomatic foundations, and practical applications with notation and worked examples.

Union and intersection in a Venn diagram

Union includes either set, while intersection keeps only the shared region.

Injective and surjective mappings

Arrows reveal whether inputs remain distinct and whether every codomain element is reached.

99 matching terms

Sets and foundations

Set

NotationA

Meaning

A well-defined collection of distinct objects considered as a single mathematical object.

When to use it

Use sets to specify collections, domains, solution spaces, events, and the underlying objects of mathematical structures.

Worked example

The set A={2,4,6} contains three even numbers.

Sets and foundations

Set element

Notationx

Meaning

An individual object contained in a set.

When to use it

Use elements when making statements about members of a collection.

Worked example

The number 4 is an element of A={2,4,6}.

Sets and foundations

Membership

Notationx∈A

Meaning

The relation stating that an object is an element of a set.

When to use it

Use membership notation to distinguish an element from a subset.

Worked example

4∈{2,4,6}.

Sets and foundations

Non-membership

Notationx∉A

Meaning

The relation stating that an object is not an element of a set.

When to use it

Use it to exclude values from a domain, event, or solution set.

Worked example

5∉{2,4,6}.

Sets and foundations

Roster notation

NotationA={a,b,c}

Meaning

A notation that defines a set by listing its elements between braces.

When to use it

Use it for finite sets whose members can be listed clearly.

Worked example

The vowel set can be written as V={a,e,i,o,u}.

Sets and foundations

Set-builder notation

Notation{x∈U:P(x)}

Meaning

A notation that defines a set by a property its elements must satisfy.

When to use it

Use it when listing every element would be impractical or impossible.

Worked example

The even integers are {n∈ℤ:n=2k for some k∈ℤ}.

Sets and foundations

Empty set

Notation∅

Meaning

The unique set containing no elements.

When to use it

Use it for an impossible event, an inconsistent solution set, or an empty intersection in context.

Caution

The empty set is a subset of every set, but it is not necessarily an element of every set.

Worked example

The real solution set of x²+1=0 is ∅.

Sets and foundations

Singleton set

Notation{x}

Meaning

A set containing exactly one element.

When to use it

Use it when a collection has one possible value or a solution is unique.

Worked example

The equation x-3=0 has the solution set {3}.

Sets and foundations

Universal set

NotationU

Meaning

The set of all objects currently under consideration.

When to use it

State it before using complements or quantifying over a fixed universe.

Caution

A universal set depends on context; it is not an absolute set of everything.

Worked example

If U=ℤ, then the complement of the even integers is the odd integers.

Sets and foundations

Set equality

NotationA=B

Meaning

Two sets are equal when they contain exactly the same elements.

When to use it

Use extensional equality without regard to element order or repetition in a written list.

Worked example

The sets {1,2,2,3} and {3,2,1} are equal.

Sets and foundations

Subset

NotationA⊆B

Meaning

A set A is a subset of B when every element of A is also an element of B.

When to use it

Use it to express containment and prove set equality by double inclusion.

Worked example

{1,3}⊆{1,2,3}.

Sets and foundations

Proper subset

NotationA⊊B

Meaning

A subset that is not equal to the containing set.

When to use it

Use it when containment is strict.

Caution

Some books use ⊂ for proper subset while others use it for any subset; define the convention.

Worked example

{1,3}⊊{1,2,3}.

Sets and foundations

Superset

NotationB⊇A

Meaning

A set that contains every element of another set.

When to use it

Use it as the reversed form of the subset relation.

Worked example

{1,2,3}⊇{1,3}.

Sets and foundations

Cardinality

Notation|A|

Meaning

A measure of the number of elements in a set.

When to use it

Use it to compare finite sizes and, through bijections, the sizes of infinite sets.

Worked example

If A={a,b,c}, then |A|=3.

Sets and foundations

Finite set

Meaning

A set whose elements can be placed in bijection with {1,...,n} for some nonnegative integer n.

When to use it

Use it when a collection has a definite whole-number size.

Worked example

The days of the week form a finite set of cardinality 7.

Sets and foundations

Infinite set

Meaning

A set that is not finite.

When to use it

Use it for unbounded collections such as integers, sequences, and points on a line.

Worked example

The integer set ℤ is infinite.

Sets and foundations

Power set

Notation𝒫(A)

Meaning

The set containing every subset of A.

When to use it

Use it to describe all possible selections, events, and binary feature combinations.

Worked example

If A={a,b}, then 𝒫(A)={∅,{a},{b},{a,b}}.

Sets and foundations

Indexed family of sets

Notation{Aᵢ}ᵢ∈I

Meaning

A collection of sets labeled by elements of an index set.

When to use it

Use it for sequences of sets and unions or intersections over arbitrary index sets.

Worked example

The family {Aₙ}ₙ∈ℕ can be defined by Aₙ={1,...,n}.

Set operations

Union

NotationA∪B

Meaning

The set of elements that belong to A, B, or both.

When to use it

Use it to combine alternatives, events, categories, or result sets.

Worked example

If A={1,2} and B={2,3}, then A∪B={1,2,3}.

Set operations

Intersection

NotationA∩B

Meaning

The set of elements that belong to both A and B.

When to use it

Use it to apply simultaneous conditions or find common members.

Worked example

If A={1,2} and B={2,3}, then A∩B={2}.

Set operations

Set difference

NotationA∖B

Meaning

The set of elements in A that are not in B.

When to use it

Use it to remove exclusions or compare what remains unique to one set.

Worked example

If A={1,2,3} and B={2,4}, then A∖B={1,3}.

Set operations

Complement

NotationAᶜ

Meaning

The set of elements in the universal set that are not in A.

When to use it

Use it for negated conditions and complementary probability events.

Worked example

If U={1,2,3,4} and A={1,3}, then Aᶜ={2,4}.

Set operations

Symmetric difference

NotationA△B

Meaning

The set of elements belonging to exactly one of A and B.

When to use it

Use it to measure disagreement between sets or toggle membership.

Worked example

If A={1,2} and B={2,3}, then A△B={1,3}.

Set operations

Disjoint sets

NotationA∩B=∅

Meaning

Sets with no elements in common.

When to use it

Use it for mutually exclusive events and nonoverlapping partitions.

Worked example

The even and odd integers are disjoint.

Set operations

Partition of a set

Meaning

A collection of nonempty pairwise disjoint subsets whose union is the original set.

When to use it

Use it to group every element into exactly one class.

Worked example

The residue classes modulo 3 partition ℤ.

Set operations

Cartesian product

NotationA×B

Meaning

The set of all ordered pairs whose first component is in A and second component is in B.

When to use it

Use it to build coordinates, relations, tables, and product spaces.

Worked example

If A={1,2} and B={x,y}, then A×B={(1,x),(1,y),(2,x),(2,y)}.

Set operations

Ordered pair

Notation(a,b)

Meaning

A pair in which the position of each component matters.

When to use it

Use it for coordinates and as the basic element of a Cartesian product or relation.

Worked example

An ordered pair usually changes when its components are swapped, so (1,2)≠(2,1).

Set operations

De Morgan's laws for sets

Notation(A∪B)ᶜ=Aᶜ∩Bᶜ

Meaning

Rules that exchange unions and intersections when taking complements.

When to use it

Use them to simplify negated set conditions and probability events.

Worked example

The second law is (A∩B)ᶜ=Aᶜ∪Bᶜ.

Set operations

Distributive laws for sets

NotationA∩(B∪C)=(A∩B)∪(A∩C)

Meaning

Rules describing how union and intersection distribute over each other.

When to use it

Use them to rewrite set expressions and prove identities.

Worked example

Also A∪(B∩C)=(A∪B)∩(A∪C).

Set operations

Absorption laws

NotationA∪(A∩B)=A

Meaning

Identities in which combining a set with a contained intersection or union returns the original set.

When to use it

Use them to remove redundant parts of set expressions.

Worked example

The dual law is A∩(A∪B)=A.

Set operations

Generalized union and intersection

Notation⋃ᵢAᵢ, ⋂ᵢAᵢ

Meaning

Union or intersection taken over an indexed family of sets.

When to use it

Use it for infinitely many sets or a variable collection of conditions.

Worked example

For Aₙ={n,n+1,...}, the intersection ⋂ₙ∈ℕAₙ is empty.

Relations and orders

Binary relation

NotationR⊆A×B

Meaning

A set of ordered pairs that specifies which elements of A are related to which elements of B.

When to use it

Use it to model comparisons, connections, database links, and function graphs.

Worked example

The relation xRy defined by x≤y is a subset of ℤ×ℤ.

Relations and orders

Domain and range of a relation

Notationdom(R), ran(R)

Meaning

The domain contains first components occurring in a relation, and the range contains occurring second components.

When to use it

Use them to determine which inputs and outputs actually participate in a relation.

Worked example

If R={(1,a),(2,a),(2,b)}, then dom(R)={1,2} and ran(R)={a,b}.

Relations and orders

Inverse relation

NotationR⁻¹

Meaning

The relation obtained by reversing every ordered pair in R.

When to use it

Use it to reverse a directional relationship.

Worked example

If R={(1,a),(2,b)}, then R⁻¹={(a,1),(b,2)}.

Relations and orders

Reflexive relation

NotationxRx

Meaning

A relation on A in which every element is related to itself.

When to use it

Use it when self-comparison must always hold, as in equality and non-strict order.

Worked example

The relation ≤ is reflexive because x≤x for every real number x.

Relations and orders

Irreflexive relation

Meaning

A relation on A in which no element is related to itself.

When to use it

Use it for strict comparisons such as less than.

Worked example

The relation < is irreflexive because x<x is always false.

Relations and orders

Symmetric relation

NotationxRy⇒yRx

Meaning

A relation whose direction can be reversed for every related pair.

When to use it

Use it for mutual relationships such as equality or sharing a property.

Worked example

The relation having the same parity is symmetric on ℤ.

Relations and orders

Antisymmetric relation

NotationxRy∧yRx⇒x=y

Meaning

A relation where two-way relatedness between distinct elements is impossible.

When to use it

Use it as an axiom of partial orders.

Caution

Antisymmetric does not mean the relation lacks symmetric pairs; equal elements may relate both ways.

Worked example

The subset relation ⊆ is antisymmetric.

Relations and orders

Asymmetric relation

NotationxRy⇒¬(yRx)

Meaning

A relation where a related pair can never also occur in the reverse direction.

When to use it

Use it for strict directed comparisons.

Worked example

The strict order < is asymmetric.

Relations and orders

Transitive relation

NotationxRy∧yRz⇒xRz

Meaning

A relation that passes through an intermediate related element.

When to use it

Use it for orders, equivalence relations, reachability, and implication chains.

Worked example

Divisibility is transitive: if a divides b and b divides c, then a divides c.

Relations and orders

Equivalence relation

Notation∼

Meaning

A relation that is reflexive, symmetric, and transitive.

When to use it

Use it to group objects that should be treated as the same under a chosen criterion.

Worked example

Congruence modulo n is an equivalence relation on ℤ.

Relations and orders

Equivalence class

Notation[x]

Meaning

The set of all elements equivalent to a given element.

When to use it

Use it as one block of the partition induced by an equivalence relation.

Worked example

For congruence modulo 3, the equivalence class of 1 is [1]={...,−5,−2,1,4,7,...}.

Relations and orders

Quotient set

NotationA/∼

Meaning

The set of all equivalence classes of A under an equivalence relation.

When to use it

Use it to replace equivalent elements with a single abstract class.

Worked example

The quotient ℤ/3ℤ has the three classes [0], [1], and [2].

Relations and orders

Partial order

Notation≼

Meaning

A reflexive, antisymmetric, and transitive relation.

When to use it

Use it when some elements are comparable while others may not be.

Worked example

Subset inclusion partially orders a power set.

Relations and orders

Partially ordered set

Notation(P,≼)

Meaning

A set together with a specified partial order.

When to use it

Use it as the object studied by order theory and dependency analysis.

Worked example

The divisors of 12 form a poset under divisibility.

Relations and orders

Total order

Notation≤

Meaning

A partial order in which every pair of elements is comparable.

When to use it

Use it for sorting and linear rankings.

Worked example

The usual order ≤ is a total order on ℝ.

Relations and orders

Hasse diagram

Meaning

A simplified graph of a finite poset that shows cover relations and omits transitive edges.

When to use it

Use it to visualize hierarchy, divisibility, subset inclusion, and dependencies.

Worked example

A Hasse diagram for the divisors of 6 places 1 below 2 and 3, with 6 above both.

Relations and orders

Well-order

Meaning

A total order in which every nonempty subset has a least element.

When to use it

Use it for induction, recursive definitions, and ordinal theory.

Worked example

The usual order on ℕ is a well-order.

Relations and orders

Minimal and maximal elements

Meaning

Elements with no strictly smaller or strictly larger comparable element in a poset.

When to use it

Use them when a partial order may have several local boundary elements.

Worked example

A finite poset can have several maximal elements.

Relations and orders

Least and greatest elements

Notation⊥, ⊤

Meaning

Elements below or above every element of a poset.

When to use it

Use them for global bounds and lattice endpoints.

Caution

Minimal does not always mean least, and maximal does not always mean greatest.

Worked example

If a least element exists, it is unique.

Relations and orders

Upper and lower bounds

Meaning

Elements that lie above or below every element of a chosen subset in an ordered set.

When to use it

Use them to define suprema, infima, bounded sets, and optimization limits.

Worked example

The number 10 is an upper bound of {1,4,7}.

Relations and orders

Supremum and infimum

Notationsup(S), inf(S)

Meaning

The least upper bound and greatest lower bound of a subset when they exist.

When to use it

Use them in analysis, optimization, and complete lattice theory.

Worked example

For S=(0,1), sup(S)=1 and inf(S)=0 even though neither belongs to S.

Functions and mappings

Function

Notationf:A→B

Meaning

A relation assigning each element of A to exactly one element of B.

When to use it

Use functions to model deterministic mappings, transformations, and computations.

Worked example

The rule f(n)=n² defines a function from ℤ to ℕ.

Functions and mappings

Domain of a function

Notationdom(f)

Meaning

The set of permitted input values of a function.

When to use it

State it because the same formula can define different functions on different domains.

Worked example

For f:ℝ→ℝ with f(x)=x², the domain is ℝ.

Functions and mappings

Codomain

NotationB

Meaning

The declared target set of a function f:A→B.

When to use it

Use it to define surjectivity and distinguish intended outputs from outputs actually attained.

Worked example

For f:ℝ→ℝ with f(x)=x², the codomain is ℝ.

Functions and mappings

Range of a function

Notationf(A)

Meaning

The set of output values actually attained by a function.

When to use it

Use it to test surjectivity and determine feasible outputs.

Caution

The range may be smaller than the codomain.

Worked example

For f:ℝ→ℝ with f(x)=x², the range is [0,∞).

Functions and mappings

Image of a subset

Notationf(S)

Meaning

The set of function values obtained from elements of a subset S of the domain.

When to use it

Use it to track how a mapping transforms a selected region or collection.

Worked example

For f(x)=x² and S={−2,1,3}, f(S)={1,4,9}.

Functions and mappings

Preimage of a subset

Notationf⁻¹(T)

Meaning

The set of domain elements whose function values lie in a chosen target subset T.

When to use it

Use it to pull conditions and events back through a function.

Worked example

For f(x)=x², the preimage of {4} is {−2,2}.

Functions and mappings

Injective function

Notationf(x₁)=f(x₂)⇒x₁=x₂

Meaning

A function that never maps two distinct inputs to the same output.

When to use it

Use it when inputs must remain distinguishable after mapping.

Worked example

The function f:ℤ→ℤ defined by f(n)=2n is injective.

Functions and mappings

Surjective function

Notationf(A)=B

Meaning

A function whose range equals its codomain.

When to use it

Use it when every declared target must be reached by at least one input.

Worked example

The function f:ℝ→[0,∞) defined by f(x)=x² is surjective.

Functions and mappings

Bijective function

NotationA↔B

Meaning

A function that is both injective and surjective.

When to use it

Use it to pair two sets element by element, compare cardinalities, and define an inverse function.

Worked example

The function f:ℤ→ℤ defined by f(n)=n+1 is bijective.

Functions and mappings

Inverse function

Notationf⁻¹:B→A

Meaning

A function that reverses a bijection by sending each output back to its unique input.

When to use it

Use it to undo a reversible mapping.

Caution

The preimage notation f⁻¹(T) is defined for subsets even when an inverse function does not exist.

Worked example

If f(x)=2x+1 on ℝ, then f⁻¹(y)=(y−1)/2.

Functions and mappings

Function composition

Notationg∘f

Meaning

A function formed by applying f first and then g.

When to use it

Use it to build complex transformations from simpler steps.

Worked example

If f(x)=x+1 and g(x)=2x, then (g∘f)(x)=2x+2.

Functions and mappings

Identity function

Notationid_A

Meaning

The function that maps every element of a set to itself.

When to use it

Use it as the neutral element for function composition.

Worked example

For every function f:A→B, f∘id_A=f and id_B∘f=f.

Functions and mappings

Restriction of a function

Notationf|_S

Meaning

A function obtained by limiting the domain of f to a subset S.

When to use it

Use it to study local behavior or make a function injective on a smaller domain.

Worked example

The square function restricted to [0,∞) is injective.

Functions and mappings

Indicator function

Notation1_A(x)

Meaning

A function that returns 1 for elements in A and 0 for elements outside A.

When to use it

Use it to encode membership algebraically in probability, integration, and data processing.

Worked example

For A={2,4}, 1_A(2)=1 and 1_A(3)=0.

Infinite sets and cardinality

Equinumerous sets

Notation|A|=|B|

Meaning

Sets connected by a bijection, meaning they have the same cardinality.

When to use it

Use bijections to compare sizes without directly counting, especially for infinite sets.

Worked example

The natural numbers ℕ and even natural numbers are equinumerous via f(n)=2n.

Infinite sets and cardinality

Countable set

Meaning

A finite set or a set that can be injected into the natural numbers.

When to use it

Use it for collections whose elements can be listed in a sequence, possibly with gaps.

Worked example

Every subset of ℕ is countable.

Infinite sets and cardinality

Countably infinite set

Notation|A|=ℵ₀

Meaning

An infinite set that can be placed in bijection with the natural numbers.

When to use it

Use it to distinguish sequence-sized infinity from larger cardinalities.

Worked example

The integer set ℤ and rational number set ℚ are countably infinite.

Infinite sets and cardinality

Uncountable set

Meaning

A set that cannot be placed in bijection with any subset of the natural numbers.

When to use it

Use it for larger infinities such as the real numbers and function spaces.

Worked example

The interval [0,1] is uncountable.

Infinite sets and cardinality

Aleph-null

Notationℵ₀

Meaning

The cardinality of the natural numbers and every countably infinite set.

When to use it

Use it as the smallest infinite cardinal.

Worked example

|ℕ|=|ℤ|=|ℚ|=ℵ₀.

Infinite sets and cardinality

Cardinality of the continuum

Notation𝔠

Meaning

The cardinality of the real numbers, equal to the cardinality of the power set of ℕ.

When to use it

Use it for the size of intervals, real-valued sequences, and continuous geometric sets.

Worked example

|ℝ|=|𝒫(ℕ)|=𝔠=2^ℵ₀.

Infinite sets and cardinality

Cantor's diagonal argument

Meaning

A method that constructs an object differing from the nth listed object at its nth component.

When to use it

Use it to prove that a proposed listing is incomplete, especially for real numbers or infinite sequences.

Worked example

Diagonalization proves that the binary sequences cannot be listed by ℕ.

Infinite sets and cardinality

Cantor's theorem

Notation|A|<|𝒫(A)|

Meaning

The power set of any set has strictly greater cardinality than the original set.

When to use it

Use it to show that there is no largest cardinal and to generate larger infinities.

Worked example

No function from A to 𝒫(A) can be surjective.

Infinite sets and cardinality

Cardinal arithmetic

Notationκ+λ, κλ, κ^λ

Meaning

Arithmetic operations defined on cardinal numbers through disjoint unions, Cartesian products, and function sets.

When to use it

Use it to compare sizes of combined infinite collections.

Worked example

For infinite countable sets, ℵ₀+ℵ₀=ℵ₀ and ℵ₀·ℵ₀=ℵ₀.

Infinite sets and cardinality

Dedekind-infinite set

Meaning

A set that is equinumerous with one of its proper subsets.

When to use it

Use it as a structural characterization of infinity in standard set theory.

Worked example

The map n↦n+1 is a bijection from ℕ to the proper subset ℕ∖{0}.

Axioms and foundations

Naive set theory

Meaning

An informal approach that treats sets as arbitrary collections described by understandable properties.

When to use it

Use it for ordinary mathematics when foundational paradoxes are not at issue.

Caution

Unrestricted collection by a property leads to paradoxes, so formal foundations use axioms.

Worked example

Basic union and intersection calculations usually need only naive set theory.

Axioms and foundations

Russell's paradox

NotationR={x:x∉x}

Meaning

The contradiction produced by asking whether the set of all sets that are not members of themselves is a member of itself.

When to use it

Use it to understand why unrestricted set comprehension is invalid.

Worked example

Assuming R∈R gives R∉R, while assuming R∉R gives R∈R.

Axioms and foundations

Axiomatic set theory

Meaning

A formal theory that permits sets and constructions only through specified axioms.

When to use it

Use it to provide a consistent foundation for mathematics and avoid known paradoxes.

Worked example

ZF and ZFC are standard axiomatic systems for set theory.

Axioms and foundations

Axiom of extensionality

Meaning

Two sets are equal exactly when they have the same elements.

When to use it

Use it to make membership completely determine the identity of a set.

Worked example

To prove A=B, it is enough to prove x∈A if and only if x∈B for every x.

Axioms and foundations

Axiom of pairing

Meaning

For any objects a and b, a set {a,b} exists.

When to use it

Use it to construct pairs and singleton sets.

Worked example

Taking a=b yields the singleton {a}.

Axioms and foundations

Axiom of union

Notation⋃A

Meaning

For any set of sets A, a set containing exactly the elements of its member sets exists.

When to use it

Use it to flatten one level of nested sets and construct unions.

Worked example

For the set A={{1,2},{2,3}}, applying the union axiom gives ⋃A={1,2,3}.

Axioms and foundations

Axiom of power set

Meaning

For every set A, a set containing exactly all subsets of A exists.

When to use it

Use it to construct function spaces, topologies, and larger cardinalities.

Worked example

The axiom guarantees the existence of 𝒫(A).

Axioms and foundations

Axiom of infinity

Meaning

An axiom asserting the existence of an inductive set that supports construction of the natural numbers.

When to use it

Use it to ensure that set theory contains at least one infinite set.

Worked example

The natural numbers can be constructed inside an inductive set.

Axioms and foundations

Axiom schema of separation

Meaning

A schema allowing elements satisfying a property to be selected from an already existing set.

When to use it

Use it to define subsets without allowing an unrestricted set of everything satisfying a property.

Worked example

Given A and property P, separation forms {x∈A:P(x)}.

Axioms and foundations

Axiom schema of replacement

Meaning

A schema stating that the image of a set under a definable functional rule is also a set.

When to use it

Use it for transfinite constructions and images indexed by large ordinals.

Worked example

A definable rule F sends a set A to the set {F(x):x∈A}.

Axioms and foundations

Axiom of foundation

Meaning

Every nonempty set contains an element disjoint from the set, preventing infinite descending membership chains.

When to use it

Use it to rule out ordinary sets such as x∈x and circular membership chains.

Worked example

Foundation excludes a two-set cycle with a∈b and b∈a.

Axioms and foundations

Axiom of choice

Meaning

For every family of nonempty sets, a function exists that chooses one element from each set.

When to use it

Use it in results such as the well-ordering theorem, Zorn's lemma, and existence of vector-space bases.

Worked example

The axiom provides a choice function even when no explicit selection rule is known.

Axioms and foundations

ZF set theory

NotationZF

Meaning

Zermelo-Fraenkel set theory without adding the axiom of choice.

When to use it

Use it as a standard formal foundation when the status of choice is kept separate.

Worked example

ZF includes extensionality, pairing, union, power set, infinity, separation, replacement, and foundation.

Axioms and foundations

ZFC set theory

NotationZFC

Meaning

ZF set theory together with the axiom of choice.

When to use it

Use it as the most common foundational framework for mainstream mathematics.

Worked example

Most ordinary mathematical results are formalizable in ZFC.

Axioms and foundations

Transitive set

Meaning

A set whose every element is also a subset of the set.

When to use it

Use it in ordinal theory, set hierarchies, and models of set theory.

Worked example

The set {∅,{∅}} is transitive.

Axioms and foundations

Ordinal number

Notationα,β,ω

Meaning

A canonical set representing the order type of a well-ordered set.

When to use it

Use it to describe positions, transfinite induction, and stages beyond finite order.

Worked example

The first infinite ordinal is ω, following all finite ordinals.

Axioms and foundations

Cardinal number

Notationκ,λ

Meaning

A canonical representative of the size shared by equinumerous sets.

When to use it

Use it to compare set sizes independently of order or internal structure.

Worked example

The finite cardinal 3 represents every three-element set.

Applications

Sample space and event

NotationΩ, E⊆Ω

Meaning

In probability, the sample space is the set of possible outcomes and an event is one of its subsets.

When to use it

Use set operations to combine events and complements to express failure.

Worked example

For a die, Ω={1,2,3,4,5,6} and the even event is E={2,4,6}.

Applications

Solution set

Meaning

The set of all values satisfying an equation, inequality, or system of constraints.

When to use it

Use it to express zero, one, several, or infinitely many solutions uniformly.

Worked example

The real solution set of x²=4 is {−2,2}.

Applications

Carrier set

Meaning

The underlying set of elements on which an algebraic or logical structure is defined.

When to use it

Use it to separate raw elements from the operations and relations added to them.

Worked example

A group (G,*) has carrier set G and operation *.

Applications

Database set operations

Meaning

Operations such as UNION, INTERSECT, and EXCEPT that combine compatible query results using set-like semantics.

When to use it

Use them to merge, compare, or subtract result rows.

Caution

Database tables can contain duplicates and null values, so SQL semantics are not identical to pure set theory.

Worked example

UNION removes duplicate rows unless UNION ALL is used.

Applications

Set data structure

Meaning

A programming collection that stores unique values and usually supports fast membership tests.

When to use it

Use it for deduplication, visited-state tracking, and membership lookup.

Worked example

A set can reduce the list [3,1,3,2] to the unique values {1,2,3}.

Applications

Type interpreted as a set

Meaning

A viewpoint in which a type is treated as the set of values allowed by that type.

When to use it

Use it to reason about validation, unions, intersections, subtypes, and exhaustive cases.

Worked example

A Boolean type can be modeled by the set {true,false}.