1 · Well-defined collections & how to write a set
A set is a well-defined collection of objects — meaning anyone, given an object, can say for certain whether it belongs or not. "The five most renowned mathematicians" is NOT well-defined (people disagree). Tap each collection below to test it.
Is it a set?
Roster form ⟷ Set-builder form
Every set can be written two ways: roster form lists every element inside braces; set-builder form states the rule the elements share. Pick an example:
Membership: ∈ and ∉
Let A = {1, 2, 3, 4, 5, 6}. Type any number to test whether it belongs to A.
2 · Empty, finite and infinite sets
A set with no elements is the empty set φ (or {}). A set is finite if you could, in principle, finish counting its elements — otherwise it's infinite. A set with exactly one element is a singleton set.
Classify each set
Counting elements — n(S)
Drag the slider to build a finite set of consecutive natural numbers starting at 1, and watch n(S) update.
3 · Equal sets, subsets & power sets
Two sets are equal when they contain exactly the same elements (order and repetition don't matter). A is a subset of B (A ⊂ B) when every element of A is also in B. If additionally A ≠ B, A is a proper subset.
Build A and B, then test A ⊂ B
Tap a number to cycle: nothing → in A → in B → in both → nothing.
Power set builder
Choose up to 4 elements. The power set P(A) is the set of all subsets of A — it always has 2ⁿ members.
4 · Nested number systems & intervals
N ⊂ Z ⊂ Q ⊂ R, and the irrationals T sit inside R but outside Q. Click a number to see exactly which nested sets contain it.
Pick a number:
Intervals as subsets of R
Set the endpoints and choose open ● / closed ● at each end.
5 · The universal set & Venn diagrams
Every discussion happens inside a universal set U — the rectangle that holds everything relevant to the problem. Subsets are drawn as circles inside it. This is a Venn diagram, named after John Venn (1834–1883).
Choose a context to see what U and its subsets could mean:
6 · Operations Lab — the Venn Machine
Universe U = {1 … 12}. Tap a number to cycle it through: only A → only B → both → outside. Then pick an operation to shade it on the diagram.
Properties worth remembering
| Commutative | A ∪ B = B ∪ A A ∩ B = B ∩ A |
| Associative | (A ∪ B) ∪ C = A ∪ (B ∪ C) (A ∩ B) ∩ C = A ∩ (B ∩ C) |
| Identity | A ∪ φ = A A ∩ U = A |
| Idempotent | A ∪ A = A A ∩ A = A |
| Law of U / φ | U ∪ A = U φ ∩ A = φ |
| Distributive | A ∩ (B ∪ C) = (A∩B) ∪ (A∩C) A ∪ (B∩C) = (A∪B) ∩ (A∪C) |
7 · Complement & De Morgan's laws
The complement A′ is everything in U that is not in A, i.e. A′ = U − A. Universe U = {1 … 10}. Tap numbers to build A.
| Complement laws | A ∪ A′ = U A ∩ A′ = φ |
| Double complement | (A′)′ = A |
| Empty / Universal | φ′ = U U′ = φ |
De Morgan's laws — see both sides match
Also build B below, then switch law. The two shaded diagrams always cover exactly the same region — that's the law.