Pearson Edexcel International GCSE Mathematics A 4MA1: Set language and notation – Revision Notes

These revision notes cover section 1.5, Set language and notation, of the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), for papers sat in January and June. Foundation statements 1.5 A–E apply to both tiers. Higher statements 1.5 A–D (algebraic sets, subsets, n(A) and practical problems) are marked Higher tier only. For full explanations and worked examples, read the set language and notation study guide.

Other pages for this section and topic: the set language practice questions, the topic 1 revision notes, the topic 1 practice questions, the Numbers and the number system study guide, the Edexcel IGCSE Mathematics hub, the printable 4MA1 checklist and the free 10-minute diagnostics.

Key definitions (both tiers)

• Set: a clearly defined collection of objects. You must be able to say, for any object, whether it belongs.
• Element (member): one object in a set.
• Universal set ℰ: everything under consideration in the question. Every other set is drawn from it.
• Empty set ∅: a set with no elements.
• Complement A′: the elements of ℰ that are not in A.

Listing rules: use curly brackets, separate elements with commas, list each element once, and order does not matter.

Notation table

Symbol	Read as	Tier
{ }	the set of	Both
∈	is an element of	Both
∉	is not an element of	Both
∩	intersection (“and”, in both)	Both
∪	union (“or”, in either or both)	Both
ℰ	the universal set	Both
∅	the empty set	Both
A′	the complement of A	Both
{x : …}	the set of all x such that …	Higher tier only
⊂	is a subset of	Higher tier only
n(A)	the number of elements in A	Higher tier only

Venn diagram regions (two sets)

Words	Notation	Regions shaded
in A and B	A ∩ B	overlap only
in A or B or both	A ∪ B	both circles
in A only	A ∩ B′	left circle minus overlap
in B only	A′ ∩ B	right circle minus overlap
in neither	(A ∪ B)′ or A′ ∩ B′	outside both circles
not in both	(A ∩ B)′	everything except overlap
not in A	A′	everything outside left circle
in exactly one	(A ∩ B′) ∪ (A′ ∩ B)	both circles minus overlap

Note the pair that is easy to confuse: (A ∪ B)′ is one region (outside both circles), while (A ∩ B)′ is three regions (everything but the overlap).

Method in steps

Listing a combined set

1. Write out ℰ in full if it is short.
2. List each named set from ℰ.
3. Do brackets first, then complements, then ∩ or ∪.
4. Check each element against the words: “in A and not in B”, and so on.

Filling a Venn diagram with elements

1. Overlap first (A ∩ B).
2. Then A only and B only.
3. Then everything left over goes outside the circles, inside the rectangle.
4. Count: the total must equal the number of elements in ℰ.

Filling a Venn diagram with numbers (Higher tier only)

1. Put the count you know for the innermost region first (the overlap, or the triple overlap for three sets).
2. Subtract outwards: A only = n(A) − n(A ∩ B).
3. If the overlap is unknown, call it x and write every region in terms of x.
4. Use the total n(ℰ) to form an equation.
5. Add all regions to check against n(ℰ).

Three sets (Higher tier only)

Start with the centre region (in all three). Next, the three “two sets only” regions: subtract the centre from each pairwise total. Then the three “one set only” regions: subtract the three regions already inside that circle from its total. Finally, “none” = n(ℰ) minus the sum of the seven regions inside the circles.

Formulae and facts to recall

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
n(A′)    = n(ℰ) − n(A)
n(neither) = n(ℰ) − n(A ∪ B)

None of these is printed on the formulae sheet; they follow from counting regions, and a Venn diagram is the safest way to apply them.

Small worked reminders

Complement of a union. ℰ = {1, 2, …, 10}, P = {1, 2, 3, 4}, Q = {3, 4, 5, 6}. P ∪ Q = {1, 2, 3, 4, 5, 6}, so (P ∪ Q)′ = {7, 8, 9, 10}.

Algebraic set (Higher tier only). {x : x is an integer, 0 ≤ x < 4} = {0, 1, 2, 3}. Strict “<” leaves out 4; “≤” keeps 0.

Subset test (Higher tier only). Is {1, 5} ⊂ {odd numbers}? Both 1 and 5 are odd, so yes. Is {1, 2} ⊂ {odd numbers}? No, because 2 is not odd. One counter-example is enough.

Counting (Higher tier only). n(A) = 15, n(B) = 11, n(A ∩ B) = 6 gives n(A ∪ B) = 15 + 11 − 6 = 20.

Unknown overlap (Higher tier only). n(ℰ) = 50, n(A) = 30, n(B) = 26 and 4 elements are in neither set.

n(A ∪ B) = 50 − 4 = 46
n(A ∩ B) = 30 + 26 − 46 = 10
A only = 20, B only = 16
Check: 20 + 10 + 16 + 4 = 50 ✓

Three-set centre first (Higher tier only). n(P ∩ Q ∩ R) = 3 and n(P ∩ Q) = 7. The region “P and Q but not R” holds 7 − 3 = 4, not 7. If n(P) = 20, n(P ∩ R) = 5 and the centre is 3, then “P and R but not Q” holds 2, and “P only” holds 20 − 3 − 4 − 2 = 11.

Must-know distinctions

• ∈ versus ⊂ (Higher tier only for ⊂). An element belongs to a set: 3 ∈ {1, 3, 5}. A set is a subset of a set: {3} ⊂ {1, 3, 5}. Writing 3 ⊂ {1, 3, 5} is wrong.
• ∩ versus ∪. ∩ is “and”, so the answer is usually smaller than either set. ∪ is “or”, so the answer is at least as big as either set.
• ∅ versus {0}. ∅ has no elements; {0} has one element, zero.
• A′ ∩ B versus (A ∩ B)′. The first is “B only”; the second is “everything except the overlap”.
• n(A) versus A. n(A) is a number; A is a set in curly brackets.
• “Only” versus total. “18 study Spanish” includes those who study both; “18 study only Spanish” does not.

Quick self-test

Questions 1–6 suit both tiers; questions 7–11 are Higher tier only.

ℰ = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3}, B = {2, 4, 6, 8}.

1. List A ∪ B.
2. List A ∩ B.
3. List A′.
4. List (A ∪ B)′.
5. Write the symbol for “is not an element of”.
6. Write the set of months whose names begin with Z, using a symbol.
7. (Higher) List {x : x is an integer, −1 < x ≤ 3}.
8. (Higher) Find n({x : x is a prime number less than 20}).
9. (Higher) n(A) = 12, n(B) = 9 and n(A ∩ B) = 4. Find n(A ∪ B).
10. (Higher) Of 30 people, 18 have a cat, 15 have a dog and 6 have neither. How many have both?
11. (Higher) Is {2, 4} ⊂ {1, 2, 3, 4}? Give a reason.

Answers

1. {1, 2, 3, 4, 6, 8}
2. {2}
3. {4, 5, 6, 7, 8}
4. {5, 7}
5. ∉
6. ∅
7. {0, 1, 2, 3}
8. The primes are 2, 3, 5, 7, 11, 13, 17, 19, so n = 8.
9. 12 + 9 − 4 = 17.
10. 30 − 6 = 24 have at least one pet; 18 + 15 − 24 = 9 have both.
11. Yes: 2 and 4 are both elements of {1, 2, 3, 4}.

Where marks are usually lost

• Listing a shared element twice in a union, which loses the accuracy mark even when the method is clear.
• Taking the complement from the wrong universal set, or forgetting that a complement needs ℰ at all.
• Writing {∅} or {0} for the empty set instead of ∅.
• Shading (A ∩ B)′ when the question asks for (A ∪ B)′, or the other way round.
• Writing the total for a set straight into its “only” region in a survey question.
• Leaving the “neither” region blank, so the diagram does not add up to n(ℰ).
• Giving n(A) as a list of elements, or giving a list when the question asks “how many”.
• Including an end point that a strict inequality excludes, such as 4 in {x : 0 ≤ x < 4}.
• Answering “is A a subset of B?” with “no” but no element to show why, which loses the reasoning mark.
• In three-set diagrams, subtracting the centre region only once from a circle’s total instead of removing all three inner regions.

Next steps

Go back to the study guide for any self-test question you got wrong, then try the full practice set. For Venn diagrams in probability, see the statistics and probability revision notes.

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification Issue 2, November 2017, Pearson Education Limited. Topic 1, Numbers and the number system: section 1.5, Set language and notation (Foundation Tier statements A–E; Higher Tier statements A–D), with the notation listed in Appendix 6.