Set notation describes the regions of a Venn diagram. A ∩ B, the intersection, means items in both A and B. A ∪ B, the union, means items in A or B or both. A′, the complement, means items not in A, and ξ is the universal set of everything. n(A) is the number of items in A, and P(A ∩ B) is the probability of being in both.
Set notation
- ξ: the universal set (everything)
- A ∩ B: in both A and B (intersection)
- A ∪ B: in A or B or both (union)
- A′: not in A (complement)
- n(A): the number of elements in A
- A′ ∩ B: in B but not in A
The symbols
The universal set ξ contains everything in the question. A ∩ B, read “A intersection B”, is the overlap. A ∪ B, “A union B”, is everything in either circle, including the overlap. A′, “A complement”, is everything outside A. One way to remember: ∩ looks like an n, for “and”, and ∪ looks like a u, for “union”.
Counting with set notation
n(A) means the number of elements in A. With A only 18, both 9, B only 20 and neither 13, out of 60: n(A ∩ B) = 9, n(A ∪ B) = 18 + 9 + 20 = 47, n(A′) = 20 + 13 = 33, and n((A ∪ B)′) = 13.
Combined regions
Combined expressions are read one symbol at a time. A′ ∩ B means in B but not in A, which is the “B only” region. (A ∩ B)′ means everything except the overlap. Shading each part of the expression separately, then finding where they agree, avoids mistakes.
Listing elements
Sets can also be listed. If ξ = {1, 2, …, 12}, A = multiples of 3 and B = even numbers, then A = {3, 6, 9, 12} and B = {2, 4, 6, 8, 10, 12}. So A ∩ B = {6, 12} and A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12}.
Probabilities with set notation
Divide the count in the region by n(ξ). With 60 people in total, P(A ∪ B) = 47/60 and P(A′) = 33/60 = 11/20. Conditional probabilities use the given set as the denominator: P(A | B) = n(A ∩ B) ÷ n(B).
Worked examples
Example 1. A Venn diagram has A only 18, A ∩ B 9, B only 20 and 13 outside both circles. Find n(A ∩ B) and n(A ∪ B).
- n(A ∩ B) is the overlap: 9.
- n(A ∪ B) = 18 + 9 + 20.
Answer: n(A ∩ B) = 9; n(A ∪ B) = 47.
Example 2. Using the same diagram, find n(A′) and n((A ∪ B)′).
- A′ is everything outside A: 20 + 13 = 33.
- (A ∪ B)′ is outside both circles: 13.
Answer: n(A′) = 33; n((A ∪ B)′) = 13.
Example 3. Using the same diagram, find P(A′).
- n(ξ) = 18 + 9 + 20 + 13 = 60.
- P(A′) = 33/60.
Answer: 11/20.
Example 4. ξ = {1, 2, …, 12}, A = {multiples of 3}, B = {even numbers}. List A ∩ B and A′ ∩ B.
- A = {3, 6, 9, 12}; B = {2, 4, 6, 8, 10, 12}.
- A ∩ B: in both. A′ ∩ B: in B but not A.
Answer: A ∩ B = {6, 12}; A′ ∩ B = {2, 4, 8, 10}.
Example 5. Using the first Venn diagram, find P(A | B).
- n(B) = 9 + 20 = 29.
- P(A | B) = n(A ∩ B) ÷ n(B).
Answer: 9/29.
Common mistakes
- Confusing ∩ and ∪. ∩ is both (and); ∪ is either (or).
- Leaving out the overlap in A ∪ B. The union includes the overlap.
- Forgetting the outside region in A′. A′ includes everything not in A, including outside both circles.
- Using the wrong denominator for P(A | B). Divide by n(B), not by n(ξ).
- Misreading combined expressions. Shade each part separately, then find where they overlap.
How set notation is examined
Set notation with Venn diagrams is Higher tier content in Edexcel GCSE (1MA1), and the International GCSE (4MA1) uses set notation at both tiers. Questions ask you to:
- complete a Venn diagram from set descriptions or listed elements
- find n(A ∩ B), n(A ∪ B) or n(A′)
- shade a region described in set notation
- find probabilities, including conditional ones, using set notation
Set notation questions, with answers, are on the Venn diagrams worksheet. For fresh diagrams to practise on, try the GCSE probability engine.
This builds on Venn diagrams, and links to conditional probability.
Frequently asked questions
What does A ∩ B mean?
The intersection: everything in both A and B.
What does A ∪ B mean?
The union: everything in A or B or both.
What does A′ mean?
The complement of A: everything that is not in A.