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.
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.
Diagram 1. Mathematical sets may contain absolutely any type of objects or entities.
Notation Used in Set Theory
Notation
Notation Used in Set Theory
The discipline that lends its marks to all the others starts here: the braces that build sets, the bar that filters them, the membership pair, and the symbol for having nothing. Sets take capital letters (A, B, C) by convention; the full chart of set theory symbols lists every mark with its LaTeX code, and the mathematical keyboard types them directly. The summary table below collects the symbols at a glance.
The containment family ⊆, ⊂, ⊇, ⊃ appears in Relationships Between Sets below and is owned by the subsets page; the operation marks ∩, ∪ belong to operations.
{1,2,3}
the set containing one, two, three
Enumerative (roster) notation: curly braces list the members outright, and membership is all they claim — order and repetition carry no meaning. Anything can be a member: numbers, letters, words, other sets.
CasesUnlistable tails compress to an ellipsis — {1,2,3,…} continues the visible pattern; ranges of functions borrow the roster for finite output lists.
Do not confuseThe other braces and brackets. A single brace selects rows in piecewise definitions; square brackets build intervals — {2,7} is two numbers, [2,7] is infinitely many.
{x∣x>0}
the set of all x such that x is positive
Descriptive (set-builder) notation: four building blocks — a variable, the vertical bar reading "such that", a condition, and the enclosing braces. It names sets too large or too awkward to list.
CasesBoth slots accept words as freely as symbols — {prime number ∣ prime number >10} is legal; domain descriptions run this grammar with the colon variant {x:…} as an alternative bar.
Do not confuseOther vertical bars. The same stroke means divides in number theory and absolute value when doubled around a quantity — between a variable and its condition, inside braces, it is always "such that".
∈ · ∈/
is an element of; is not an element of
The membership pair: x∈A claims x is a member, and the slash manufactures the denial — y∈/B. The slash-negation pattern is general equipment: =, ∈/, ⊂ all deny by striking through.
CasesMembership doubles as a domain declaration across mathematics — the n∈Z of general solutions and the z∈C of complex numbers are this mark at work abroad.
Do not confuseContainment. 1∈A but {1}⊆A — an element belongs, a set is contained; mixing ∈ with ⊆ is the subject's most reliable exam error, and Relationships Between Sets below draws the line.
∅
the empty set
The set with no members, written ∅ or {} — one object, two spellings. It is a set like any other: a legitimate value, a legitimate member, a legitimate answer.
CasesNesting is not emptiness: {∅} is a one-element set whose single member happens to be empty — ∣∅∣=0 but ∣{∅}∣=1.
Do not confuseThe Greek letter phi. Handwriting collapses ∅ into ϕ — but the empty-set mark is a struck circle, not a letter; writing "phi" for the empty set is a naming error graders notice.
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.
Pitfall: Counting a repeated element twice
Taking ∣{1,2,2,3}∣=4 counts the entry 2 twice. The listing names only the elements 1, 2 and 3, so the size is 3, the same as for {1,2,3}.
To find a cardinality from a listing, read off the members one by one and count them. The order in which they are written plays no part, so {3, 1, 2} gives the same count as {1, 2, 3}, and a member that is itself a word, a number or a symbol counts in the same way as any other. For a finite set the counting always comes to an end, and its result is the cardinality: 0 for the empty set and a positive whole number for any other finite 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 counting its elements would eventually end, even if only in theory, 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.
Pitfall: Putting the empty set inside braces
Writing the empty set as {∅} gives a set with one element, the empty set itself, so ∣{∅}∣=1. The empty set is ∅ or {} on its own, with ∣∅∣=0.
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
A set is finite when its elements can be counted to an end and infinite when they cannot. The universal set is the frame for everything under consideration.
L0Types of sets
L1Finitea well-defined number of elements{1, 2, 3, 4, 5}
L2Empty seta special case of a finite set∅
L1Infinitecannot be exhausted by counting{1, 2, 3, …}
L1Universalall objects under consideration; finite or infiniteU
Takeaway
Size is cardinality
How to compare the sizes of sets, including infinite ones, is the subject of cardinality.
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.
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; for a finite set, it contains fewer elements. Example: A = {1, 2} is a proper subset of B.
Pitfall: Using the membership sign for a subset
Writing {1,2}∈B for B={1,2,3,4} claims that the set {1,2} is one of the four elements of B, and it is not. The correct statements are {1,2}⊆B and 1∈B.
The example set B = {1, 2, 3, 4} has many subsets besides A = {1, 2}. The one-element sets such as {3} are subsets of B, and so are the empty set and B itself, since each of them is built only from elements of B. Their sizes run from 0 elements for the empty set up to 4 elements for B itself, and only B is a subset of B that is not proper. In all, a set with 4 elements has 24=16 subsets.
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.
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.
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 Ac = {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}
UAB
A ⊆ B: every element of A is an element of B
The circle A lies entirely inside the circle B, so any element that belongs to A automatically belongs to B: that containment is the subset relation, and the picture shows why B can have elements outside A while the reverse is impossible. Membership of a single point and containment of a whole set are the two relationships drawn here. Drag the circles apart and watch the relation change on the two-set Venn diagram explorer.
Equality of sets is the case where the containment runs both ways.
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}.
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}.
Pitfall: Adding the sizes for a union
Taking 3+3=6 as the size of A∪B for A={1,2,3} and B={4,5,3} counts the shared element 3 twice. The union {1,2,3,4,5} has 5 elements: 3+3−1=5.
The shared element 3 belongs to A and to B at the same time, and the union takes it in like any other element. Union is the set version of the word "or": an element qualifies as soon as it belongs to at least one of the two sets, and belonging to both is allowed. The elements 1 and 2 come into the union from A alone, and 4 and 5 from B alone. The diagram below shades both circles completely, the overlap included.
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}.
Pitfall: Reversing the difference
Computing B−A in place of A−B for A={1,2,3,4} and B={3,4,5,6} gives {5,6}, the part of B outside A. Order matters: A−B={1,2} and B−A={5,6} are different sets.
The difference starts from the first set and removes whatever the second set also contains. In the example, A loses 3 and 4 because B contains them, and it keeps 1 and 2, which B lacks. The elements 5 and 6 belong to B alone, so they play no part in the result: the difference only ever takes elements away from A and never adds any. The diagram below shades the part of the circle A that lies outside 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}.
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}.
UAB
A ∩ B: the shaded overlap
Only the region belonging to both circles is shaded: an element is in A ∩ B exactly when it is in A and in B. The union would shade both circles entirely, the difference A \ B only the part of A outside the overlap. Switch operations and watch the shading move on the two-set Venn diagram explorer.
The operations page develops each of these with its own laws and diagrams.
Common Pitfalls
Five mistakes come up again and again with the basic vocabulary of sets. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.
Counting a repeated element twice
A set is written down with one element listed twice, and its size is read off as the number of entries in the listing. Repeated entries name the same element, so they add nothing to the set.
Wrong working: {1,2,2,3} has four entries, so ∣{1,2,2,3}∣=4.
Right working: the distinct elements are 1, 2 and 3, so {1,2,2,3}={1,2,3} and ∣{1,2,2,3}∣=3.
Why it happens: counting the entries on the page is quicker than checking each one against the others, and many lists outside mathematics, such as shopping lists or tallies, do count repeats.
How to check: rewrite the set with each element once, in any order, before counting. The number of entries in the rewritten listing is the cardinality.
Putting the empty set inside braces
The empty set is written as {∅}, with its symbol placed inside a pair of braces. The braces build a new set whose one element is the empty set, so the result is no longer empty.
Wrong working: E={∅} is taken to be the empty set, so ∣E∣=0.
Right working: {∅} has exactly one element, so ∣{∅}∣=1. The empty set is written ∅ or {} on its own, and ∣∅∣=0.
Why it happens: other sets are written with braces, so wrapping the symbol in braces looks like proper set notation. But ∅ is already a complete name for a set and needs no braces around it.
How to check: ask what is listed between the outer braces. In {∅} there is one item, so the set is not empty.
Using the membership sign for a subset
The sign ∈ is written between two sets when one is contained in the other. ∈ says that its left side is one of the elements of the right side; containment of a whole set is written ⊆.
Wrong working: for B={1,2,3,4}, the statements {1,2}∈B and 1⊆B are written.
Right working: the elements of B are the numbers 1, 2, 3 and 4, and the set {1,2} is not one of them, so {1,2}∈/B but {1,2}⊆B. The number 1 is an element, so 1∈B, and the one-element set built from it is a subset: {1}⊆B.
Why it happens: both signs are often read aloud as "is in", and in everyday speech a group of items can be in a collection just as a single item can.
How to check: read x∈B as "x appears in the listing of B". Look for {1,2}, braces included, among the listed elements of B; it is not there.
Adding the sizes for a union
The size of a union is found by adding the sizes of the two sets. Elements that belong to both sets are then counted once for each set.
Wrong working: for A={1,2,3} and B={4,5,3}, the size of A∪B is given as 3+3=6.
Right working: A∪B={1,2,3,4,5}, which has 5 elements. In general the size of the union is ∣A∣+∣B∣−∣A∩B∣, and here A∩B={3}, so 3+3−1=5. The inclusion-exclusion principle extends this correction to more than two sets.
Why it happens: the union is built by putting two sets together, and putting together sounds like adding.
How to check: write the union out and count it, or test the answer against its bounds. The union has at least as many elements as the larger set and at most ∣A∣+∣B∣, and it reaches ∣A∣+∣B∣ only when the sets are disjoint.
Reversing the difference
A−B is computed as if the order of the two sets did not matter, and the elements of B outside A are returned. Unlike union and intersection, the difference can change when the sets swap places.
Wrong working: for A={1,2,3,4} and B={3,4,5,6}, the difference is given as A−B={5,6}.
Right working: start from A and remove the elements that are also in B, namely 3 and 4. This leaves A−B={1,2}. The set {5,6} is the other difference, B−A.
Why it happens: union, intersection and symmetric difference give the same set in either order, and the habit carries over to the one basic operation where order matters.
How to check: every element of A−B must be an element of A. Since 5 and 6 are not in A, the set {5,6} cannot be A−B.
Set Theory Basics FAQ
What can be an element of a set?
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Anything at all. Elements need not share a type or have anything in common, so {1, 'apple', 3.14, true} is a perfectly valid set of four elements. Sets can even contain other sets. The only rules are that elements are distinct and unordered, which is why {1, 2, 2, 3} and {3, 1, 2} name the same set.Read more →
What is the difference between roster and set-builder notation?
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Roster notation lists the members outright: {2, 4, 6}. Set-builder states a rule instead: {x | x is a positive even number}, read as the set of all x such that x is even. Roster suits small finite sets; set-builder handles sets too large or awkward to write out. Both describe the same kind of object. See all set theory symbols.Read more →
What is the universal set?
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The universal set U contains every object under consideration in a given context. It is not a fixed collection: for a weekday problem U has seven elements, and for a real-analysis problem U is all of the reals. Complements depend entirely on it, since the complement of A means everything in U outside A.Read more →
Is the empty set the same as a set containing the empty set?
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No. The empty set has no members and cardinality 0, written as an empty pair of braces. The set {∅} has exactly one member, which happens to be the empty set, so its cardinality is 1. Think of an empty box versus a box holding an empty box. Both are legitimate sets.Read more →
Are {1, 2, 3} and {3, 2, 1, 1} the same set?
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Yes. Sets are unordered, so rearranging the listing changes nothing, and elements are distinct, so repeating one adds nothing. Both expressions name the same three-element set. This is also why cardinality counts distinct members only: the second listing has four entries written down but still has size 3.
