A ∪ B, frozen
Set Operations › Union
A ∩ B, frozen
Set Operations › Intersection
A′, frozen
Set Operations › Complement
A ∖ B, frozen
Set Operations › Set Difference
A △ B, frozen
Set Operations › Symmetric Difference
A = B, frozen
Set Relationships › Equal Sets
Pairing off, and failing to
Set Relationships › Equivalent Sets
A ∩ B = ∅, frozen
Set Relationships › Disjoint Sets
Overlapping sets, frozen
Set Relationships › Overlapping Sets
Three blocks that fill the set
Set Relationships › Partitions
A ⊆ B, frozen
Subsets and Power Sets › Subsets
Three elements, three choices, eight subsets
Subsets and Power Sets › Number of Subsets
𝒫({a, b, c}), frozen
Subsets and Power Sets › Power Set
The bare anatomy
Venn Diagrams › What Are Venn Diagrams
The four regions, labeled
Venn Diagrams › Two-Set Venn Diagrams
A ∩ B ∩ C, frozen
Venn Diagrams › Three-Set Venn Diagrams
A ∪ (B ∩ C), frozen
Venn Diagrams › Shading Regions
(A ∪ B)ᶜ, frozen
Venn Diagrams › Using Venn Diagrams to Verify Set Identities
Why the correction term exists
Cardinality › Finite Sets
The rule n ↦ 2n, on an infinite set and on a finite one
Cardinality › Infinite Sets
Listing the positive rationals
Cardinality › Countable Sets
Cantor's diagonal, five rows deep
Cardinality › Uncountable Sets
A ⊆ B: every element of A is an element of B
Set Theory Basics: Terminology and Core Concepts › Relationships Between Sets
A ∩ B: the shaded overlap
Set Theory Basics: Terminology and Core Concepts › Operations on Sets
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), frozen
Set Theory Rules › Distributive Laws
(A ∪ B)′ = A′ ∩ B′, frozen
Set Theory Rules › De Morgan’s Laws
Before any term, frozen
Inclusion-Exclusion Principle Explorer › Nothing Counted Yet
Every overlap zero, frozen
Inclusion-Exclusion Principle Explorer › When Nothing Overlaps
Sizes that cannot happen, frozen
Inclusion-Exclusion Principle Explorer › Sizes That Cannot Happen
|A| and |B| applied, frozen
Inclusion-Exclusion Principle Explorer › A Term Being Added
All three sets in, frozen
Inclusion-Exclusion Principle Explorer › All the Sets Are In
Subtracting |A ∩ B|, frozen
Inclusion-Exclusion Principle Explorer › A Term Being Subtracted
All three subtractions done, frozen
Inclusion-Exclusion Principle Explorer › The Centre Falls to Zero
All seven terms applied, frozen
Inclusion-Exclusion Principle Explorer › Every Region Counted Once
All terms applied, still not resolved
Inclusion-Exclusion Principle Explorer › When the Walk-Through Does Not Resolve
Elements box empty, frozen
Power Set Explorer › An Empty Elements Box
a, b, a — frozen
Power Set Explorer › A Repeated Element
Five elements, frozen at the cap
Power Set Explorer › Past Five Elements
A one-element set, frozen
Power Set Explorer › A One-Element Set
Three elements, nothing selected
Power Set Explorer › The Whole Lattice, Nothing Selected
The empty set selected, frozen
Power Set Explorer › The Empty Set at the Bottom
The whole set selected, frozen
Power Set Explorer › The Whole Set at the Top
A singleton selected, frozen
Power Set Explorer › A Singleton Selected
A pair inside a four-element set, frozen
Power Set Explorer › A Subset in the Middle
No condition after the bar, frozen
Set-Builder Notation Explorer › An Empty Condition
−4 against a domain of ℕ, frozen
Set-Builder Notation Explorer › A Number Outside the Domain
x < 4 over the naturals, frozen
Set-Builder Notation Explorer › The Ordinary Bounded Case
Even and below twelve, frozen
Set-Builder Notation Explorer › Two Conditions Joined by And
Even or below twelve, frozen
Set-Builder Notation Explorer › Two Conditions Joined by Or
7 tested against the primes, frozen
Set-Builder Notation Explorer › A Candidate That Passes
9 tested against the primes, frozen
Set-Builder Notation Explorer › A Candidate That Fails
Even and odd at once, frozen
Set-Builder Notation Explorer › Nothing Passes the Filter
Prime and even, frozen
Set-Builder Notation Explorer › Exactly One Survivor
The even naturals, frozen
Set-Builder Notation Explorer › A Set Too Big to List
Set A, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Set A on Its Own
Set B, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Set B on Its Own
Set C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Set C on Its Own
Universal set U, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Universal Set U
Empty set ∅, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Empty Set
Complement A′, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Complement of A
Complement B′, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Complement of B
Complement C′, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Complement of C
A ∩ B ∩ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Triple Intersection
A ∩ B, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Pairwise Intersection A ∩ B
A ∩ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Pairwise Intersection A ∩ C
B ∩ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Pairwise Intersection B ∩ C
A ∪ B ∪ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Triple Union
A ∪ B, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Pairwise Union A ∪ B
A ∪ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Pairwise Union A ∪ C
B ∪ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Pairwise Union B ∪ C
A ∖ B, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Difference A Minus B
A ∖ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Difference A Minus C
B ∖ A, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Difference B Minus A
B ∖ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Difference B Minus C
C ∖ A, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Difference C Minus A
C ∖ B, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Difference C Minus B
A ∖ (B ∪ C), frozen
Three-Set Venn Diagram Basic Identities Explorer › Only in A
B ∖ (A ∪ C), frozen
Three-Set Venn Diagram Basic Identities Explorer › Only in B
C ∖ (A ∪ B), frozen
Three-Set Venn Diagram Basic Identities Explorer › Only in C
(A ∪ B) ∖ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Union of A and B Minus C
A △ B, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Symmetric Difference of A and B
A △ B △ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Triple Symmetric Difference
A ∩ B ∩ C′, frozen
Three-Set Venn Diagram Basic Identities Explorer › In A and B but Not C
A ∩ B′ ∩ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › In A and C but Not B
A′ ∩ B ∩ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › In B and C but Not A
A ∩ (B ∪ C), frozen
Three-Set Venn Diagram Basic Identities Explorer › A Intersected with B Union C
(A ∪ B) ∩ C, frozen
Three-Set Venn Diagram Basic Identities Explorer › A Union B Intersected with C
A ∪ (B ∩ C), frozen
Three-Set Venn Diagram Basic Identities Explorer › A United with B Intersect C
Exactly one, frozen
Three-Set Venn Diagram Basic Identities Explorer › Exactly One of the Three Sets
Exactly two, frozen
Three-Set Venn Diagram Basic Identities Explorer › Exactly Two of the Three Sets
At least two, frozen
Three-Set Venn Diagram Basic Identities Explorer › At Least Two of the Three Sets
At most one, frozen
Three-Set Venn Diagram Basic Identities Explorer › At Most One of the Three Sets
(A ∪ B ∪ C)′, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Complement of the Triple Union
(A ∩ B ∩ C)′, frozen
Three-Set Venn Diagram Basic Identities Explorer › The Complement of the Triple Intersection
Set A, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Set A on Its Own
Set B, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Set B on Its Own
Universal set U, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Universal Set U
Empty set ∅, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Empty Set
Complement A′, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Complement of A
Complement B′, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Complement of B
Intersection A ∩ B, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Intersection of A and B
Union A ∪ B, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Union of A and B
Difference A ∖ B, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Difference A Minus B
Difference B ∖ A, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Difference B Minus A
Symmetric difference A △ B, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Symmetric Difference
A ∪ B′, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Union of A with the Complement of B
A′ ∪ B, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Union of the Complement of A with B
(A ∪ B)′, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Complement of the Union
(A ∩ B)′, frozen
Two-Set Venn Diagram Basic Identities Explorer › The Complement of the Intersection
A ⊆ B, frozen
Two-Set Venn Diagram Basic Identities Explorer › A as a Subset of B
B ⊆ A, frozen
Two-Set Venn Diagram Basic Identities Explorer › B as a Subset of A
Disjoint sets, frozen
Two-Set Venn Diagram Basic Identities Explorer › Disjoint Sets
Equal sets, frozen
Two-Set Venn Diagram Basic Identities Explorer › Equal Sets
A ∪ A = A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Idempotent Law for Union
A ∩ A = A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Idempotent Law for Intersection
A ∪ B = B ∪ A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Commutative Law for Union
A ∩ B = B ∩ A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Commutative Law for Intersection
A ∪ ∅ = A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Union with the Empty Set
A ∩ U = A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Intersection with the Universe
A ∩ ∅ = ∅, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Intersection with the Empty Set
A ∪ U = U, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Union with the Universe
A ∪ A′ = U, frozen
Two-Set Venn Diagram Laws and Identities Explorer › A Set United with Its Complement
A ∩ A′ = ∅, frozen
Two-Set Venn Diagram Laws and Identities Explorer › A Set Intersected with Its Complement
(A′)′ = A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Double Complement
U′ = ∅, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Complement of the Universe
∅′ = U, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Complement of the Empty Set
(A ∪ B)′ = A′ ∩ B′, frozen
Two-Set Venn Diagram Laws and Identities Explorer › De Morgan's Law for the Union
(A ∩ B)′ = A′ ∪ B′, frozen
Two-Set Venn Diagram Laws and Identities Explorer › De Morgan's Law for the Intersection
A ∪ (A ∩ B) = A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Absorption by Union
A ∩ (A ∪ B) = A, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Absorption by Intersection
A ∖ B = A ∩ B′, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Difference as Intersection
B ∖ A = A′ ∩ B, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Reversed Difference
A △ B = (A ∖ B) ∪ (B ∖ A), frozen
Two-Set Venn Diagram Laws and Identities Explorer › Symmetric Difference from Two Differences
A △ B = (A ∪ B) ∩ (A ∩ B)′, frozen
Two-Set Venn Diagram Laws and Identities Explorer › Symmetric Difference as Union Minus Intersection
(A △ B)′ = (A ∩ B) ∪ (A ∪ B)′, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Complement of the Symmetric Difference
(A′ ∪ B)′ = A ∩ B′, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Complement of A′ ∪ B
(A ∪ B′)′ = A′ ∩ B, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Complement of A ∪ B′
(A ∩ B′)′ = A′ ∪ B, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Complement of A ∩ B′
(A′ ∩ B)′ = A ∪ B′, frozen
Two-Set Venn Diagram Laws and Identities Explorer › The Complement of A′ ∩ B
(A ∪ B) ∪ C = A ∪ (B ∪ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › Associativity of Union
(A ∩ B) ∩ C = A ∩ (B ∩ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › Associativity of Intersection
(A △ B) △ C = A △ (B △ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › Associativity of the Symmetric Difference
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › Intersection Distributes over Union
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › Union Distributes over Intersection
(A ∪ B ∪ C)′ = A′ ∩ B′ ∩ C′, frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › The Complement of the Triple Union
(A ∩ B ∩ C)′ = A′ ∪ B′ ∪ C′, frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › The Complement of the Triple Intersection
A ∖ (B ∪ C) = (A ∖ B) ∩ (A ∖ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › Difference over a Union
A ∖ (B ∩ C) = (A ∖ B) ∪ (A ∖ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › Difference over an Intersection
(A ∪ B) ∖ C = (A ∖ C) ∪ (B ∖ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › A Union Minus a Set
(A ∩ B) ∖ C = A ∩ (B ∖ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › An Intersection Minus a Set
(A ∖ B) ∖ C = A ∖ (B ∪ C), frozen
Three-Set Venn Diagram Laws and Complex Identities Explorer › The Nested Difference
Two sets: A ∩ B, A ∪ B, A ⊕ B, Aᶜ
Venn Diagram Generator › The Four Operations on Two Sets
A ∩ B under the Disjoint and A ⊆ B layouts
Venn Diagram Generator › Layouts Change Which Regions Exist
Three sets: A ∩ B ∩ C, then (A ∪ B)ᶜ
Venn Diagram Generator › Three Sets and De Morgan's Law
Four sets (A ∩ B) and five sets (A ∩ B ∩ C)
Venn Diagram Generator › Four and Five Sets
A single index, frozen
Indexed Union and Intersection Explorer › One Set Is Not a Family
Three terms in, frozen
Indexed Union and Intersection Explorer › The Pattern Is Still Forming
Multiples of the index, frozen
Indexed Union and Intersection Explorer › The Running Intersection Is an LCM
Three sets on a Venn diagram, frozen
Indexed Union and Intersection Explorer › The Finite Chain
Sliding tails, frozen
Indexed Union and Intersection Explorer › The General Case
[−i, i] at five terms, frozen
Indexed Union and Intersection Explorer › Bounded Sets, Unbounded Union
A downward-nested family, frozen
Indexed Union and Intersection Explorer › When the Union Stops Growing
An upward-nested family, frozen
Indexed Union and Intersection Explorer › When the Intersection Stops Shrinking
(0, 1/i] at six terms, frozen
Indexed Union and Intersection Explorer › An Empty Intersection of Non-Empty Sets
[0, 1/i] at six terms, frozen
Indexed Union and Intersection Explorer › When One Bracket Decides the Answer
No expression yet, frozen
Venn Diagram and Truth Table Explorer › An Empty Expression Box
A ∩ (B ∪ C) on three sets, frozen
Venn Diagram and Truth Table Explorer › An Ordinary Mixed Column
A ∩ B with the A-only row picked, frozen
Venn Diagram and Truth Table Explorer › One Row, One Region
A ∪ (B — frozen
Venn Diagram and Truth Table Explorer › An Expression That Will Not Parse
A ∩ C in two-set mode, frozen
Venn Diagram and Truth Table Explorer › A Set That Is Not on the Diagram
A ∪ Aᶜ, frozen
Venn Diagram and Truth Table Explorer › Every Region Shaded
A ∩ Aᶜ, frozen
Venn Diagram and Truth Table Explorer › No Region Shaded
Aᶜ ∪ B, frozen
Venn Diagram and Truth Table Explorer › When the Column Has a Name
A ∩ (A ∪ B), frozen
Venn Diagram and Truth Table Explorer › A Column That Ignores a Set