Venn diagram symbols and set notation
What ∪, ∩, A′ and the rest mean, with a shading calculator that shows the region for any expression you type.
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Symbol reference
| Symbol | Name | Say it | Meaning |
|---|---|---|---|
| ∪ | Union | A union B | Everything in A or B (or both). |
| ∩ | Intersection | A intersect B | Only what is in both A and B. |
| A′ or Aᶜ | Complement | not A, A prime | Everything in the universal set that is not in A. |
| − or \ | Difference | A minus B | In A but not in B. Same as A ∩ B′. |
| Δ | Symmetric difference | A delta B | In exactly one of A or B, not both. |
| U or ξ | Universal set | the universe | Everything being considered; drawn as the rectangle. |
| ∅ or { } | Empty set | the empty set | A set with no elements. A ∩ B = ∅ means A and B don’t overlap. |
| ∈ | Element of | is in | 3 ∈ A means 3 is an element of A. |
| ∉ | Not an element of | is not in | 7 ∉ A means 7 is not in A. |
| ⊆ | Subset | is a subset of | Every element of A is also in B; A’s circle sits inside B’s. |
| ⊂ | Proper subset | is a proper subset of | A ⊆ B and B has at least one element A doesn’t. |
| n(A) or |A| | Cardinality | the number of elements in A | How many elements a set has, e.g. n(A ∩ B) = 4. |
Union and intersection, the two you’ll use most
Think of ∪ as “or” and ∩ as “and”. If A is students who play football and B is students who play tennis, A ∪ B is everyone who plays at least one, and A ∩ B is the students who play both.
The two are linked by the addition rule: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). You subtract the overlap because it was counted once in A and again in B. The Venn diagram calculator lets you drag through that idea, and the practice questions put the notation to work.
De Morgan’s laws
(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Type each side into the shading calculator: the same region lights up both times, which is the easiest way to remember them.
Shading checklist for exams
- Draw the universal set rectangle if the question mentions U, ξ or a complement.
- Work inside brackets first.
- Shade lightly for each part, then keep only what the operator asks for: both (∩), either (∪).
- Check with a quick element: pick a point in each region and ask “is it in the expression?”
Questions people ask
What does ∩ mean in a Venn diagram?
∩ means intersection: the region where the circles overlap. A ∩ B contains only the elements that are in both A and B.
What does ∪ mean in a Venn diagram?
∪ means union: everything inside either circle, including the overlap. A ∪ B contains the elements in A, in B, or in both.
What does A′ (A prime) mean?
A′ is the complement of A: everything inside the universal set rectangle that is outside circle A. It is also written Aᶜ or Ā.
How do you shade A ∩ B′?
Shade the part of A that is not in B, the left crescent of a two-circle diagram. It’s the same region as A − B. Type it into the shading calculator above to see it.
What is the symbol for the universal set?
Usually U, and in UK schools often ξ (xi). It is drawn as the rectangle around all the circles.
What is the difference between ⊂ and ⊆?
A ⊆ B allows A and B to be equal. A ⊂ B (proper subset) means A is inside B and B has something extra. Many textbooks use ⊂ for both, so check your course’s convention.