Sheet № 241 · Higher only · AQA · Edexcel · OCR
Set Notation and Venn Diagrams –
Set notation and Venn diagrams are Higher-tier topics that appear on every GCSE exam board — AQA, Edexcel and OCR. You need to understand the formal symbols for union, intersection and complement, read and shade Venn diagrams, and use them to solve probability problems. These skills are worth several marks and are often combined with othe
§Key definitions
Question:
ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {2, 4, 6, 8, 10}. B = {3, 6, 9}. (a) Draw a Venn diagram. (b) List A ∩ B. (c) List A'.
(b)
A ∩ B = {6}.
(c)
A' = everything not in A = {1, 3, 5, 7, 9}.
Answer:
(a) Venn diagram with 6 in the overlap, {2,4,8,10} in A only, {3,9} in B only, {1,5,7} outside. (b) {6}. (c) {1, 3, 5, 7, 9}.
(a)
n(R ∪ C) = 20 + 8 + 14 = 42. P(R ∪ C) = 42/50 = 21/25.
§Formulas to memorise
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
P(A) = n(A) ÷ n(ξ)
A ∪ B — (A union B) — everything in A or B or both.
A ∩ B — (A intersection B) — everything in both A and B.
A' — (A complement) — everything not in A (but in the universal set).
ξ — (xi) — the universal set containing all elements under consideration.
n(A) — the number of elements in set A.
∈ — "is a member of" (e.g. 3 ∈ A means 3 is in set A).
∅ — the empty set (a set with no elements).
A ⊂ B — A is a subset of B (every element of A is also in B).
Worked example
ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {2, 4, 6, 8, 10}. B = {3, 6, 9}. (a) Draw a Venn diagram. (b) List A ∩ B. (c) List A'.
Working:
⚠ Common mistakes
- ✗Confusing ∪ and ∩. Union (∪) means "or" and includes everything in either set. Intersection (∩) means "and" and only includes what is in both.
- ✗Counting the intersection twice. When calculating n(A ∪ B), remember to subtract n(A ∩ B) once: n(A) + n(B) − n(A ∩ B).
- ✗Forgetting the "neither" region. Always subtract the total in all circles from n(ξ) to find how many elements are outside all sets.
- ✗Misreading complement notation. A' means everything in ξ that is NOT in A, not just B.
✦ Exam tips
- →Always start by filling in the intersection region first — everything else follows from that.
- →When a question uses set notation like (A ∪ B)', shade the diagram to visualise what is being asked.
- →Check that all regions of your Venn diagram add up to n(ξ).
- →For basic Venn diagram questions without formal notation, see Venn diagrams. For conditional probability extensions, see conditional probability.