Set notation and Venn diagrams
The symbols, and using Venn diagrams to solve problems.
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 1.1.1Use set language, notation and Venn diagrams to describe sets and solve problems.
6 minute read
Set notation and Venn diagrams
Set notation is used far more heavily in 0607 than in 0580, and the symbols must be read fluently.
The notation
A ∪ B— union: everything in A or B (or both).A ∩ B— intersection: everything in both A and B.A'— complement: everything not in A.∈— "is an element of";∉— "is not an element of".⊂— "is a subset of".n(A)— the number of elements in A.∅or{ }— the empty set.ℰ— the universal set, everything under consideration.
A memory hook that works: ∪nion looks like a cup that holds everything; ∩ntersection is the overlap.
Reading combined expressions
Work from the inside out, exactly like arithmetic brackets:
(A ∪ B)'— everything outside both circles.A ∩ B'— in A but not in B, i.e. the part of A that does not overlap.n(A ∪ B) = n(A) + n(B) − n(A ∩ B)— subtract the overlap, otherwise it is counted twice.
Solving Venn diagram problems
The reliable method is to start in the middle:
- Fill in
n(A ∩ B)first. - Subtract it from each total to get the parts that are only in A and only in B.
- Fill in the outside region last, using the universal set total.
Starting from the outside almost always leads to double counting.
Why it matters
Set notation also underpins probability questions, where P(A ∩ B) and P(A ∪ B) mean exactly the same thing as in the diagrams.
Think of it like this
A Venn diagram is a seating plan: people in the overlap belong to both clubs and would be counted twice if you simply added the club memberships — which is precisely why the intersection is subtracted.
Worked examples
Method, step by step
In a class of 30, 18 study French and 15 study Spanish, and 7 study both. How many study neither?
- 1Start with the intersection: 7 study both.
- 2French only = 18 − 7 = 11; Spanish only = 15 − 7 = 8.
- 3Total studying at least one = 11 + 7 + 8 = 26.
- 4Neither = 30 − 26 = 4.
4 students study neither
Common misconceptions
- Confusing ∪ and ∩. Union is the larger set (everything in either); intersection is the smaller (only the overlap).
- Adding `n(A) + n(B)` for the union without subtracting the intersection, which double-counts the middle.
- Filling a Venn diagram from the outside in, rather than starting with the intersection.
In the exam
- Always write the middle value first, then work outwards subtracting as you go.
- Shade the region described by an expression before counting — for something like `A ∩ B'`, shading removes most of the ambiguity.