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Set Theory Basics: Terminology and Core Concepts



Key Terms
What Is a Set? The Definition
Notation Used in Set Theory
Cardinality and Types of Sets
Relationships Between Sets
Operations on Sets



The Vocabulary Every Other Page Builds On


Set theory is the language in which much of modern mathematics is written. This page collects its basic vocabulary in one place: what a set is, how sets are written down, how their size is measured, and the standard relationships and operations between sets. Every topic surveyed here is developed in depth on its own dedicated page within the set theory section.

Key Terms

Set— an unordered collection of unique items
Element— a single member of a set
Empty Set— the set containing no elements
Universal Set— the set of everything under consideration
Subset— a set whose elements all belong to another set
Union— the elements found in either of two sets
Intersection— the elements common to both sets
Cardinality— the number of elements in a set

See All Set Theory Definitions →


What Is a Set? The Definition


    In mathematics, a set is defined as an unordered collection of unique items. This short yet concise definition contains everything needed to understand the essence of the object we call a mathematical set. Let us analyze it in more depth:

  • Unordered — the order of the elements does not matter. Two or more sets containing the same items are identical, no matter how the items are listed.

  • Collection — stands for zero or more items, up to an infinite number of them.

  • Unique — each member of the set is distinct; no two identical items are present.

    • What kinds of items or entities can be members of a mathematical set? Whatever they are, it does not matter. Mathematics is all about abstraction, and in the abstract world things can be of any type — so the famous 'apples and oranges' problem does not apply here. Entities of completely different kinds can live together as parts of the same set.
"A" 1
Diagram 1. Mathematical sets may contain absolutely any type of objects or entities.

Notation Used in Set Theory


    It is a widespread convention to denote mathematical sets by capital English letters (for instance AA, BB, CC). Although there are no strict rules here, following the convention helps maintain standards of communication in the academic world.

    Enumerative Set Notation

    This method is straightforward: it lists the elements of a set within curly braces. There are several ways to describe a mathematical set using enumerative notation.

    Listing all elements of a set within curly braces:

  • AA = {1, 2, 3}

  • BB = {a, b, c, d} — not only numbers are allowed

  • CC = {1, a, 30, 'cfgd', π} — a reminder that a set can hold items of different types

  • DD = {23, a, 😊, π, A, 🔺, 'hello', ψ} — a mathematical set can include practically anything

  • EE = { } — the empty set

    • Listing the elements we can list, then using an ellipsis (...) which stands for the omitted items:

  • AA = {1, 2, 3, ...}

  • BB = {a, b, c, d, ...}

  • CC = {1, a, b, 4, 7, ...}

    • We will see later in which cases this form is useful — for the moment, the examples themselves are enough.

    Descriptive Set Notation

    Descriptive (set-builder) notation defines a set by specifying a property that its elements must satisfy. It is particularly useful when enumerative notation would not work — when a set is too large, or its elements are difficult to list directly.

    For example, the set of all positive even numbers can be expressed as AA = {x | x is a positive even number}, where the vertical bar means "such that". This notation represents large or infinite sets without listing all their elements explicitly.

    You can think of descriptive set notation as comprising four main building blocks:

    1. Variable (element of the set) — may be denoted by symbols (x, y, z) or simply by words ({number | number is a positive even number}, {prime number | prime number > 10}).

    2. Vertical bar (|) — represents the phrase "such that", widely used in mathematics (in other words, "where" or "for which"). It always comes between the variable and the condition that follows.

    3. Condition(s) — the property or rule that defines which elements belong to the set. It must be present, has to be clear and precise, and cannot include logical contradictions. Like variables, conditions may be expressed with mathematical symbols (x > 0, or 0 ≤ y ≤ 3) or with words (x is divisible by 2, for example).

    4. Curly braces ({}) — enclose the entire notation, indicating that the expression inside defines a set. They turn everything described above into a formal mathematical expression.

    More Basic Notations Used in Set Theory

  • Empty set (∅ or {}) — represents a mathematical set which contains no elements.

  • Belongs to (∈) — x ∈ A means that x is an element of set A.

  • Does not belong to (∉) — y ∉ B means that y is not an element of set B.

    • Symbols for Concepts Covered Further Down This Page

  • Subset (⊆)

  • Proper subset (⊂)

  • Superset (⊇)

  • Proper superset (⊃)

    • These relations are discussed in the Relationships Between Sets section below.

Set Notation Summary

Concept Symbol/Notation Explanation
Enumerative Set Notation {1, 2, 3}, {a, b, c} Lists all elements of a set within curly braces.
Enumerative Set with Ellipsis {1, 2, 3, ...}, {a, b, c, d, ...} Lists some elements and uses '...' to indicate continuation, useful for large or infinite sets.
Descriptive Set Notation {x | x is a positive even number} Defines a set by specifying a condition that elements must satisfy.
Empty Set ∅, {} Represents a set with no elements.
Belongs to ∈ 'x ∈ A' means 'x' is an element of set 'A'.
Does not belong to ∉ 'y ∉ B' means 'y' is not an element of set 'B'.
Subset ⊆ 'A ⊆ B' means 'A' is a subset of 'B', meaning all elements of 'A' are in 'B'.
Proper Subset ⊂ 'A ⊂ B' means 'A' is a proper subset of 'B', meaning 'A' is a subset of 'B' but not equal to 'B'.
Superset ⊇ 'B ⊇ A' means 'B' is a superset of 'A', meaning all elements of 'A' are in 'B'.
Proper Superset ⊃ 'B ⊃ A' means 'B' is a proper superset of 'A', meaning 'B' contains all elements of 'A' but is not equal to 'A'.

Cardinality and Types of Sets


    The cardinality of a mathematical set is simply its size — the number of elements in the set. Cardinality is most often denoted by the symbol |A| or card(A), where A is the set; symbols like #A, n(A), or ‖A‖ may be used as well.

    If A = {1, 2, 3}, then the cardinality of A is |A| = 3, because there are three elements in the set. If B = {1, 'apple', 3.14, True}, then |B| = 4, because there are four elements in this set.

    Based on cardinality, mathematical sets are classified into groups:

  • Finite sets — sets having a well-defined number of elements that may be expressed by a natural number. Examples: A = {1, 2, 3, 4, 5}, B = {1, 'apple', 3.14, True}. A set may contain millions of objects, but as long as it can be counted even theoretically, the set is finite.

  • Infinite sets — sets whose elements cannot be exhausted by counting. Examples: A = {x | x is a positive even number} — it is impossible to count all positive even numbers; B = {1, 2, 3, ...}.

  • Empty set — a special case of a finite set having no elements. Notations used for the empty set: E = { } or E = ∅.

  • Universal set — a set that contains all the objects or elements under consideration within a specific context or domain of discourse. It is typically denoted U or ξ. The universal set may be finite — if it represents the days of the week, then U = {Monday, Tuesday, Wednesday, ..., Sunday}, a finite set of 7 elements. It may also be infinite — if it represents all natural numbers, then U = {1, 2, 3, ...}; if it represents all real numbers, then U = ℝ, an infinite set with an uncountable number of elements.

    • Interestingly, despite being conceptually opposite, the universal set and the empty set share fundamental commonalities in set theory. Both are extremes in the hierarchy of sets — the universal set containing everything and the empty set containing nothing. Yet they are complementary and serve as identity elements in key set operations.

    Understanding the universal set and the empty set has immense importance for mastering other ideas in set theory, such as operations on sets, relations, and more. This section only measures size by counting — the story of comparing infinite sizes continues on the dedicated cardinality page.

Cardinality and Types of Sets Summary

Concept Notation Explanation Examples
Cardinality |A|, card(A) The size or number of elements in a set |A| = 3 for A = {1, 2, 3}, |B| = 4 for B = {1, 'apple', 3.14, True}
Finite Sets Regular Set Notation (enumerative or descriptive) Sets with a countable number of elements {Monday, ..., Sunday}, {A, ..., Z}
Infinite Sets Regular Set Notation (enumerative or descriptive) Sets with an uncountable number of elements {x | x is a positive even number}, {1, 2, 3, ...}
Empty Set ∅ or {} A set containing no elements ∅, {}
Universal Set U or ξ Set containing all objects within a specific context U = {Monday, ..., Sunday} for days of the week, U = {A, ..., Z} for English alphabet

Relationships Between Sets


    Basic Set Membership

  • Belongs to (∈) — an object belongs to a set if it is an element of the set. Notation: x ∈ A means that x is an element of set A. Example: if A = {1, 2, 3}, then 1 ∈ A but 4 ∉ A.

  • Does not belong to (∉) — notation: y ∉ B means that y is not an element of set B. Example: if B = {5, 6, 7}, then 4 ∉ B.
A 1 B C 3 D 1 ∈ A B ∈ A C ∈ A 3 ∉ A D ∉ A
Diagram 2. Set membership.

    Equality and Inequality of Sets

  • Equality (=) — two sets are equal if they contain exactly the same elements. Example: A = {1, 2, 3} and B = {3, 1, 2}; A = B.

  • Inequality (≠) — two sets are unequal if there is at least one element that is different between the sets. Example: A = {1, 2, 3} and C = {4, 5, 6}; A ≠ C.

    • Subsets

  • Basic subsets — a set is a subset of another set if every element of this set is also an element of the including set. Example: A = {1, 2} is a subset of B = {1, 2, 3, 4}.

  • Proper subsets — a proper subset of a set is a subset that is not equal to the original set, containing fewer elements. Example: A = {1, 2} is a proper subset of B.

    • Supersets

  • Basic supersets — a superset includes every element of another set. Example: A = {1, 2, 3, 4} is a superset of B = {1, 2} and of C = {1, 2, 3, 4}.

  • Proper supersets — a proper superset contains all elements of the subset plus additional elements. Example: A is a proper superset of B, but not a proper superset of C.
U B A
Diagram 3. Set A is a subset of set B, and set B is a superset of set A.

Disjoint Sets

Two sets are disjoint if they have no common elements. Example: A = {1, 2, 3} and C = {4, 5, 6} are disjoint.
U A B
Diagram 4. Disjoint sets have no common area on a Venn diagram.

Complement

The complement of set A with respect to a universal set U includes all elements that are in U but not in A. Example: U = {1, 2, 3, 4, 5}; A = {1, 2, 3}; then AcA^c = {4, 5}.

This section starts by establishing the fundamental notion of set membership, which is crucial for understanding all subsequent relations and operations between sets. That foundation is then expanded with the various types of set relations, each illustrated with a clear example. The full treatment of these relations lives on the relationships between sets page.

Relationships Between Sets Summary

Concept Notation Explanation Examples
Equality of Sets = Two sets are equal if they contain exactly the same elements. A = {1, 2, 3}, B = {3, 1, 2}; A = B
Inequality of Sets ≠ Two sets are unequal if at least one element is different between the sets. A = {1, 2, 3}, C = {4, 5, 6}; A ≠ C
Basic Subsets ⊆ A subset is a set where every element of the subset is also an element of another set. A = {1, 2}, B = {1, 2, 3, 4}; A ⊆ B
Proper Subsets ⊂ A proper subset of a set is not equal to the original set, containing fewer elements. A = {1, 2}, B = {1, 2, 3, 4}; A ⊂ B
Basic Supersets ⊇ A superset includes every element of another set. A = {1, 2, 3, 4}, B = {1, 2}; A ⊇ B
Proper Supersets ⊃ A proper superset contains all elements of the subset plus additional elements. A = {1, 2, 3, 4}, B = {1, 2}; A ⊃ B
Disjoint Sets N/A Two sets are disjoint if they have no elements in common. A = {1, 2, 3}, C = {4, 5, 6}; A and C are disjoint
Complement Ac The set of all elements not in the given set, relative to a universal set. U = {1, 2, 3, 4, 5}, A = {1, 2, 3}; Ac = {4, 5}

Operations on Sets


    The diagrams accompanying each operation below are Venn diagrams — the standard way to picture how sets combine.

  • Intersection (∩) — the intersection of two sets A and B, denoted A ∩ B, is the set of elements that are common to both A and B. Example: if A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∩ B = {3, 4}.
A B A∩B
Diagram 5. The intersection of two sets is their common area on a Venn diagram.

  • Union (∪) — the union of two sets A and B, denoted A ∪ B, is the set of elements that are in A, in B, or in both. Example: if A = {1, 2, 3} and B = {4, 5, 3}, then A ∪ B = {1, 2, 3, 4, 5}.
A B A∪B
Diagram 6. The union of two sets is all the area the two circles occupy together.

  • Difference (−) — the difference between two sets A and B, denoted A − B, is the set of elements that are in A but not in B. Example: if A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A − B = {1, 2}.
A B A - B
Diagram 7. The difference A − B is the part of A that does not belong to B.

  • Symmetric difference (Δ) — the symmetric difference between two sets A and B, denoted A Δ B, is the set of elements that are in either of the sets A or B but not in their intersection. Example: if A = {1, 2, 3} and B = {3, 4, 5}, then A Δ B = {1, 2, 4, 5}.
A B A Δ B
Diagram 8. The symmetric difference is the area belonging to exactly one of the two sets.

This section demonstrates the fundamental operations on sets with graphical examples to aid in understanding intersection, union, difference, and symmetric difference. The dedicated operations on sets page develops each operation in depth, together with the algebraic laws they obey.

Operations on Sets Summary

Concept Notation Explanation Examples
Intersection A ∩ B The set of elements that are common to both sets A and B. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∩ B = {3, 4}.
Union A ∪ B The set of all elements that are in either set A or set B, or in both. If A = {1, 2, 3} and B = {4, 5, 3}, then A ∪ B = {1, 2, 3, 4, 5}.
Difference A - B The set of elements that are in set A but not in set B. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A - B = {1, 2}.
Symmetric Difference A Δ B The set of elements that are in either of the sets A or B, but not in both. If A = {1, 2, 3} and B = {3, 4, 5}, then A Δ B = {1, 2, 4, 5}.