Pearson Edexcel International GCSE Mathematics A 4MA1: Set language and notation – Study Guide

This study guide teaches section 1.5, Set language and notation, of the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), first assessed in June 2018 with papers available in January and June. Section 1.5 sits in topic 1, Numbers and the number system. Foundation statements 1.5 A–E are needed for both tiers; the Higher tier adds statements 1.5 A–D, marked Higher tier only below.

Each tier sits two 2-hour papers of 100 marks (1F and 2F, or 1H and 2H), each worth 50% of the qualification. A calculator may be used on every paper, but set questions are about careful listing and counting, not calculation.

For a condensed version, see the revision notes. To test yourself, use the practice questions. The whole of topic 1 is summarised in the Numbers and the number system study guide, the topic 1 revision notes and the topic 1 practice questions. Course links: the Edexcel IGCSE Mathematics hub, the printable 4MA1 checklist and the free 10-minute diagnostics.

What this section covers

Spec ref	What you must be able to do	Tier
1.5 A	Understand the definition of a set	Both tiers
1.5 B	Use the set notation ∪, ∩, ∈ and ∉	Both tiers
1.5 C	Understand the universal set and the empty set, and their symbols ℰ and ∅	Both tiers
1.5 D	Understand and use the complement of a set, written A′	Both tiers
1.5 E	Use Venn diagrams to represent sets	Both tiers
1.5 A	Understand sets defined in algebraic terms; understand and use subsets (A ⊂ B)	Higher tier only
1.5 B	Use Venn diagrams to represent sets and the number of elements in sets	Higher tier only
1.5 C	Use the notation n(A) for the number of elements in the set A	Higher tier only
1.5 D	Use sets in practical situations	Higher tier only

The specification’s notation appendix lists the symbols used in the papers: { } (the set of), n(A), {x : } (the set of all x such that), ∈, ∉, ∅, ℰ, ∪, ∩, ⊂ and A′. Learn exactly these. The universal set symbol is printed as a curly capital E.

1.5 A: What a set is (both tiers)

A set is a collection of objects, called elements (or members). The collection must be clearly defined, so that for any object you can say whether it is in the set or not. “Even numbers less than 10” is a set. “Large numbers” is not, because nobody agrees which numbers count as large.

You write a set inside curly brackets, either by listing its elements or by describing them:

• A = {2, 4, 6, 8}
• A = {even numbers less than 10}

Two facts about listing:

• Order does not matter. {2, 4, 6, 8} and {8, 6, 4, 2} are the same set.
• Each element is listed once. The letters of the word LETTER form the set {L, E, T, R}, not {L, E, T, T, E, R}.

A set can be finite (it has a fixed number of elements) or infinite, such as {odd numbers}. In exam questions, the sets you list are finite because a universal set or a stated range limits them.

1.5 B and C: Notation, universal set and empty set (both tiers)

Symbol	Meaning	Example
∈	is an element of	4 ∈ {2, 4, 6}
∉	is not an element of	5 ∉ {2, 4, 6}
∩	intersection: elements in both sets	{1, 2, 3} ∩ {2, 3, 4} = {2, 3}
∪	union: elements in either set, or both	{1, 2, 3} ∪ {2, 3, 4} = {1, 2, 3, 4}
ℰ	the universal set: everything being considered in the question	ℰ = {1, 2, 3, …, 12}
∅	the empty set: a set with no elements	{odd multiples of 4} = ∅

A useful way to remember the two main symbols: ∩ looks like an “n” for intersection and means “and”; ∪ looks like a “U” for union and means “or”.

The universal set sets the limits of the question. If ℰ = {1, 2, 3, …, 12}, then “multiples of 3” means only 3, 6, 9 and 12, not 15 or 300.

The empty set is written ∅, with no curly brackets. Writing {∅} or { 0 } is wrong: {0} is a set with one element, the number 0.

Worked example 1. ℰ = {1, 2, 3, …, 12}. A = {multiples of 3}, B = {factors of 12}.

Step 1: list each set from ℰ
A = {3, 6, 9, 12}
B = {1, 2, 3, 4, 6, 12}

Step 2: intersection - elements in A AND in B
A ∩ B = {3, 6, 12}

Step 3: union - every element in A OR B, each listed once
A ∪ B = {1, 2, 3, 4, 6, 9, 12}

Step 4: membership
9 ∈ A   (9 is a multiple of 3)
9 ∉ B   (9 is not a factor of 12)

Notice that 3, 6 and 12 appear in both lists, but appear only once in A ∪ B.

Empty set check. With the same ℰ, let C = {odd multiples of 4}. Every multiple of 4 is even, so C = ∅, and A ∩ C = ∅.

1.5 D: The complement of a set (both tiers)

The complement of A, written A′, is the set of elements of ℰ that are not in A. You cannot find a complement without knowing ℰ.

Complements combine with ∪ and ∩. Work from the inside of any brackets outwards.

Worked example 2. Using ℰ, A and B from worked example 1, list A′, (A ∪ B)′, A′ ∩ B and A ∩ B′.

A′ = everything in ℰ not in A
   = {1, 2, 4, 5, 7, 8, 10, 11}

(A ∪ B)′: first A ∪ B = {1, 2, 3, 4, 6, 9, 12}
          then take what is left of ℰ
   = {5, 7, 8, 10, 11}

A′ ∩ B: elements NOT in A but in B
   = {1, 2, 4}

A ∩ B′: elements in A but NOT in B
   = {9}

Read A′ ∩ B aloud as “not in A, and in B”. That is the same as “in B only”.

1.5 E: Venn diagrams (both tiers)

A Venn diagram shows ℰ as a rectangle and each set as a circle (or oval) inside it. Two overlapping circles split the rectangle into four regions:

Region	Words	Set notation
Overlap of the circles	in A and in B	A ∩ B
Left circle, outside the overlap	in A only	A ∩ B′
Right circle, outside the overlap	in B only	A′ ∩ B
Inside the rectangle, outside both circles	in neither	(A ∪ B)′, also written A′ ∩ B′

Larger regions are made by joining these: A is the left circle (two regions), A ∪ B is both circles together (three regions), and A′ is everything outside the left circle.

Method for placing elements.

1. Fill the overlap first: list A ∩ B.
2. Fill “A only” and “B only”.
3. Put every remaining element of ℰ outside both circles.
4. Check: each element of ℰ appears exactly once on the diagram.

For worked examples 1 and 2 the regions are: overlap {3, 6, 12}; A only {9}; B only {1, 2, 4}; outside both {5, 7, 8, 10, 11}. That is 3 + 1 + 3 + 5 = 12 elements, which matches ℰ.

Worked example 3 (identifying a shaded region). A diagram shows everything shaded except the overlap of A and B. Write the shaded region in set notation.

Unshaded region = A ∩ B
Shaded region   = everything else in ℰ
                = (A ∩ B)′

If instead only the right circle is shaded, apart from the overlap, the region is A′ ∩ B.

Venn diagrams also appear in probability questions; see the statistics and probability study guide for that use.

1.5 A: Sets in algebraic terms and subsets (Higher tier only)

A set can be defined by a rule using the notation {x : …}, read “the set of all x such that …”.

Worked example 4.

(a) List {x : x is an integer, −3 < x ≤ 2}.

−3 is excluded (strict <), 2 is included (≤)
{−2, −1, 0, 1, 2}

(b) ℰ = {x : x is an integer, 1 ≤ x ≤ 20}. S = {x : x is a square number}, T = {x : 2x + 3 < 15}. List S and T, and decide whether S ⊂ T.

S = {1, 4, 9, 16}            (squares up to 20)

2x + 3 < 15
2x < 12
x < 6
T = {1, 2, 3, 4, 5}          (integers in ℰ less than 6)

Is S ⊂ T?  9 ∈ S but 9 ∉ T, so S is not a subset of T.

Subsets. A ⊂ B means “A is a subset of B”: every element of A is also an element of B. If A ⊂ B, the circle for A sits wholly inside the circle for B on a Venn diagram. {4, 16} ⊂ S above, because both 4 and 16 are in S. To show a set is not a subset, give one element that is in the first set but not the second.

1.5 B and C: Counting elements with n(A) (Higher tier only)

n(A) is the number of elements in A. It is a number, not a set, so you write n(A) = 4, never n(A) = {1, 4, 9, 16}.

On a Venn diagram, you can write the number of elements in each region instead of the elements themselves. Numbers in regions are added to give n of any combined set.

For two sets, the overlap is counted in both n(A) and n(B), so:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Using S and T from worked example 4: n(S) = 4, n(T) = 5, S ∩ T = {1, 4} so n(S ∩ T) = 2, and n(S ∪ T) = 4 + 5 − 2 = 7. Listing S ∪ T = {1, 2, 3, 4, 5, 9, 16} confirms 7.

1.5 D: Sets in practical situations (Higher tier only)

Practical questions give you counts from a survey and ask for a count you are not told. Always draw the Venn diagram and fill it from the inside out.

Worked example 5. In a class of 40 students, 23 study French (F), 18 study Spanish (S) and 7 study neither. How many study both? Find n(F ∩ S′).

Students in at least one: n(F ∪ S) = 40 − 7 = 33
Both: n(F ∩ S) = 23 + 18 − 33 = 8

Regions: F only = 23 − 8 = 15
         S only = 18 − 8 = 10
         neither = 7
Check: 15 + 8 + 10 + 7 = 40 ✓

n(F ∩ S′) = F only = 15

Worked example 6 (with algebra). A club has 60 members. 35 play tennis (T) and 30 play golf (G). The number who play neither is half the number who play both. Find how many play both.

Let n(T ∩ G) = x
T only = 35 − x,  G only = 30 − x
neither = 60 − (35 − x) − x − (30 − x) = x − 5

x − 5 = x/2
2x − 10 = x
x = 10

So 10 play both and 5 play neither. Check: 25 + 10 + 20 + 5 = 60 ✓.

Common errors

• Listing a repeated element twice in a union: {3, 6, 12} must not become {3, 3, 6, 6, 12, 12}.
• Finding A′ from the wrong universal set, for example including 13 when ℰ stops at 12.
• Writing the empty set as {∅} or {0}. Write ∅ on its own.
• Reading A′ ∩ B as “not in A or B”. It means “not in A, and in B”.
• Treating ⊂ and ∈ as the same: 4 ∈ S, but {4} ⊂ S. An element belongs (∈); a set is contained (⊂).
• Giving n(A) as a list instead of a number.
• In survey questions, writing the given total for A (say 23) in the “A only” region before subtracting the overlap.
• Missing the “neither” region, so the regions add up to less than n(ℰ).
• With inequalities such as −3 < x ≤ 2, including the excluded end point or dropping the included one.

Next steps

Condense this with the set language revision notes, then work through the set language practice questions. Check your coverage of the rest of the course with the 4MA1 checklist.

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.