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IGCSE Maths · Edexcel 4XMA1 · Unit 1 Higher · Number

Sets, Notation and Venn Diagrams

Sets turn up on every Edexcel Unit 1 paper, usually as a Venn diagram question worth 3–5 marks. Learn the symbols, learn to fill a Venn diagram from the inside out, and these are some of the most reliable marks on the paper.

1Sets and the symbols you must know

A set is a collection of objects, called elements or members. We write a set in curly brackets, either by listing the elements or by describing them.

SymbolMeaningExample
{ }a setA = {2, 4, 6, 8}
∈is an element of6 ∈ A
∉is not an element of5 ∉ A
ξthe universal set (everything being considered)ξ = {1, 2, 3, …, 10}
∅ or { }the empty set (no elements){odd numbers in A} = ∅
n(A)the number of elements in An(A) = 4

A set can also be described algebraically. Read the colon as “such that”:

B = {x : x is an integer, 2 < x ≤ 7}

means “B is the set of all x such that x is an integer and 2 < x ≤ 7”, so B = {3, 4, 5, 6, 7} and n(B) = 5.

Key ideaWatch the inequality signs. 2 < x does not include 2, but x ≤ 7 does include 7.

2Union, intersection and complement

These three operations combine sets. Each one matches a region of a Venn diagram.

ξAB
A ∩ B — the intersection: in A and in B
ξAB
A ∪ B — the union: in A or B (or both)
ξAB
A′ — the complement of A: not in A
ξAB
A ∩ B′ — in A but not in B
Memory trick∩ looks like an “n” — think and. ∪ looks like a cup that holds everything from both sets. The dash in A′ means not.
Worked example
ξ = {1, 2, 3, …, 10}, A = {even numbers}, B = {multiples of 3}. List (a) A ∩ B (b) A ∪ B (c) A′ ∩ B.
  1. A = {2, 4, 6, 8, 10} and B = {3, 6, 9}.
  2. (a) Elements in both: A ∩ B = {6}.
  3. (b) Elements in either, each listed once: A ∪ B = {2, 3, 4, 6, 8, 9, 10}.
  4. (c) A′ = {1, 3, 5, 7, 9}. Of these, the ones also in B: A′ ∩ B = {3, 9}.
A ∩ B = {6}; A ∪ B = {2, 3, 4, 6, 8, 9, 10}; A′ ∩ B = {3, 9}

3Subsets

If every element of B is also in A, then B is a subset of A, written B ⊂ A. On a Venn diagram, B is drawn completely inside A.

For example, if A = {1, 2, 3, 4, 5} and B = {2, 4}, then B ⊂ A. But C = {2, 6} is not a subset of A, because 6 ∉ A, so we write C ⊄ A.

Exam tipIf a question asks you to explain why B ⊂ A, say: “every element of B is also an element of A”. If it isn’t a subset, name an element of B that is not in A.

4Venn diagrams with numbers: n(A)

Often the diagram shows how many elements are in each region rather than listing them. The golden rule is to fill from the inside out: start with the overlap, then subtract it to find the “only” regions.

Worked example
In a class of 40 students, 23 play football, 18 play tennis and 7 play neither. How many play both?
  1. Students who play at least one sport: 40 − 7 = 33.
  2. Adding the two groups counts the “both” students twice: 23 + 18 = 41.
  3. The extra 41 − 33 = 8 is the overlap, so 8 play both.
  4. Fill the diagram: football only = 23 − 8 = 15, tennis only = 18 − 8 = 10, neither = 7. Check: 15 + 8 + 10 + 7 = 40 ✓
8 students play both
ξFT158107
Completed diagram for the worked example
Useful formulan(A ∪ B) = n(A) + n(B) − n(A ∩ B)

5Three-set Venn diagrams

With three sets there are eight regions. Use the same inside-out rule: centre first, then the three “pairs only” regions, then each “one set only” region, and finally the outside.

Worked example
60 people were asked which of Apple (A), Banana (B) and Cherry (C) they like. 4 like all three. 10 like A and B, 9 like A and C, 7 like B and C. 25 like A, 22 like B, 20 like C. How many like none?
  1. Centre: A ∩ B ∩ C = 4.
  2. A and B only = 10 − 4 = 6; A and C only = 9 − 4 = 5; B and C only = 7 − 4 = 3.
  3. A only = 25 − 6 − 5 − 4 = 10; B only = 22 − 6 − 3 − 4 = 9; C only = 20 − 5 − 3 − 4 = 8.
  4. Total inside the circles = 10 + 9 + 8 + 6 + 5 + 3 + 4 = 45, so none = 60 − 45 = 15.
15 people like none of the three
ξABC1098653415
Completed three-set diagram

6Probability from a Venn diagram

Once the diagram is complete, probability is just region ÷ total. Always use n(ξ) as the denominator unless the question says “given that”.

Worked example
Using the football/tennis diagram above, a student is chosen at random. Find (a) P(plays tennis only) (b) P(F ∪ T) (c) P(plays tennis, given that they play football).
  1. (a) Tennis only = 10, so P = 1040 = 14.
  2. (b) F ∪ T = 15 + 8 + 10 = 33, so P = 3340.
  3. (c) “Given that they play football” shrinks the total to the 23 football players. Of those, 8 also play tennis: P = 823.
(a) 1/4 (b) 33/40 (c) 8/23
Exam tip“Given that” changes the denominator to the size of the condition set. It’s the most common place to lose a mark on Venn probability.

7Venn diagrams with algebra

Harder questions put expressions in the regions. Add every region, set it equal to n(ξ), and solve.

Worked example
n(ξ) = 50. On a Venn diagram, A only = 3x, A ∩ B = x, B only = 2x + 2 and outside = 6. Find n(B).
  1. All regions add to 50: 3x + x + (2x + 2) + 6 = 50.
  2. Simplify: 6x + 8 = 50, so 6x = 42 and x = 7.
  3. n(B) = x + (2x + 2) = 3x + 2 = 3(7) + 2 = 23.
  4. Check: A only = 21, both = 7, B only = 16, outside = 6, total 50 ✓
n(B) = 23
ξAB3xx2x + 26
Key ideaA count of people or objects must be a whole number. If your x comes out as a fraction, re-check your equation before moving on.
Common mistakes
  • Putting the total for a set (e.g. 23 for football) in the “only” region instead of subtracting the overlap first.
  • Forgetting the region outside the circles: it is part of ξ and is needed for any probability.
  • Listing an element twice in a union, or forgetting that A ∪ B includes the overlap.
  • Misreading algebraic set notation: {x : 2 < x ≤ 7} starts at 3, not 2.
  • Using n(ξ) as the denominator for a “given that” probability.

★Key Vocabulary Recap

Set
A collection of distinct objects (elements).
Universal set ξ
The set containing every element under consideration; the rectangle in a Venn diagram.
Empty set ∅
A set with no elements.
Intersection A ∩ B
Elements in both A and B.
Union A ∪ B
Elements in A or B or both.
Complement A′
Elements of ξ that are not in A.
Subset B ⊂ A
Every element of B is also in A.
n(A)
The number of elements in set A.
Specification reference (Pearson Edexcel 4XMA1, Unit 1 Higher)
  • Understand the definition of a set; use the notation ∪, ∩, ∈ and ∉.
  • Understand the universal set ξ and the empty set ∅.
  • Understand and use the complement of a set, A′.
  • Use Venn diagrams to represent sets and the number of elements in sets; use n(A).
  • Understand sets defined in algebraic terms; understand and use subsets.
  • Use sets in practical situations; find probabilities from a Venn diagram.
Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.
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