Sets and Venn diagram
Grab a number and move it around the diagram — the membership table and the element counts are recalculated instantly.
A ∩ B · A ∪ B · A \ B
|A ∪ B| = |A| + |B| − |A ∩ B|
A ∩ B = intersection — elements belonging to A and to B
A ∪ B = union — elements belonging to A or to B
A \ B = difference — elements of A that are not in B
|A| = cardinality — the number of elements of set A
U = universe — all the elements under consideration
A set is a collection of mutually distinguishable elements. For every element we can say unambiguously whether it belongs to the set (x ∈ A) or does not belong to it (x ∉ A). A Venn diagram shows the sets as circles inside a rectangle that stands for the universe U — all the elements under consideration.
The overlap of the circles is the intersection A ∩ B, the whole area of both circles is the union A ∪ B and the part of circle A outside B is the difference A \ B. Since the common elements would be counted twice in the plain sum |A| + |B|, we subtract them once when counting the union — that is the inclusion–exclusion principle.
| Element | x ∈ A | x ∈ B | A ∩ B | A ∪ B | A \ B | B \ A |
|---|
Drag the numbers into the circles + add elementreset
- Intersection A ∩ B: elements belonging at the same time to A and to B — the overlap of the circles.
- Union A ∪ B: elements belonging to at least one of the sets; the common ones are counted only once.
- Disjoint sets: if they share no element, then A ∩ B = ∅.
- Both operations are commutative: A ∩ B = B ∩ A and A ∪ B = B ∪ A as well.
- A \ B: elements that are in A but are not in B.
- It is not commutative: A \ B and B \ A are usually different sets.
- Decomposition: A ∪ B can be split into three non-overlapping parts — A \ B, A ∩ B and B \ A.
- If A ⊆ B (A is a subset of B), then A \ B = ∅.
- Cardinality |A| is the number of elements of set A.
- Inclusion–exclusion principle: |A ∪ B| = |A| + |B| − |A ∩ B|.
- Why minus: the common elements would be counted twice in the sum |A| + |B|.
- If the sets are disjoint, the formula simplifies to |A ∪ B| = |A| + |B|.
Solve it step by step — the next step unlocks only after a correct answer. You can type decimals with a dot or a comma.
What is the intersection of sets?
What is the union of sets?
How does A \ B differ from B \ A?
Why is the intersection subtracted in the formula?
