Edexcel IGCSE

Set language and notation

A set is a collection of things, and set notation is the shorthand for saying which things. It is section 1.5 of the Edexcel International GCSE Mathematics A specification, and almost every question on it reduces to the same move: decide exactly which members of the universal set each named set contains, then read off the region the question asked about.

Edexcel IGCSE 4MA132 approved practice questions

Set up your class — free Practise on your own

No card needed. Free for individual students and teachers.

Before you start

What you need to be able to do already

If any of these is shaky, fix it first — every worked example below leans on all three.

List the members of a set from a rule — multiples, factors, primes, and the integers between two limits
Write out every multiple of $3$ up to $12$ , and then every factor of $12$ . If ${3, 6, 9, 12}$ and ${1, 2, 3, 4, 6, 12}$ did not come quickly, fix that first: sets on this page are almost never handed to you as a list, they are described by a rule and you build the list yourself.
Read an inequality as the numbers it allowsInequalities
Which integers satisfy $- 3 \leq x \leq 3$ ? All seven of $- 3, - 2, - 1, 0, 1, 2, 3$ , because $\leq$ includes both ends. Higher-tier questions hide exactly this inside set-builder brackets, and a strict $<$ instead would have dropped two of them.
Write one part of a total as a fraction, and find a fraction of a totalFractions
$7$ of $30$ students play neither sport — what fraction is that? $\frac{7}{30}$ . Venn questions rarely stop at the count: the follow-up asks for that region as a fraction or a probability, and that is where the last mark sits.

The idea

What is set language and notation?

A set is a collection of objects that is well defined — the rule is sharp enough that anyone can say whether a given thing belongs. The objects in it are its elements, they are written once each inside curly brackets, and the order does not matter, so ${3, 5, 7}$ and ${7, 5, 3}$ are the same set. ‘The prime numbers less than $10$ ’ is a set; ‘the best footballers in the world’ is not, because it is an opinion and two people would write different lists. Everything else on this page is bookkeeping on top of that one idea.

The notation, and how to read it

Every worked solution below is written in these five symbols.

SymbolReads asMeans
$ξ$ the universal setEverything under discussion in this question, and nothing outside it. It is the rectangle a Venn diagram is drawn inside. Edexcel prints it as a script $E$ , and you will also meet $U$ for the same thing. It is never the empty set.
$\in$ is an element of $5 \in A$ says $5$ is one of the members of $A$ . Struck through, $\in /$ says it is not — and true-or-false questions turn on which of the two is printed.
$\cup$ union $A \cup B$ is everything in $A$ , or in $B$ , or in both — the ‘at least one’ set. The cup opens upward and scoops both circles up.
$\cap$ intersection $A \cap B$ is only what is in both sets at once — the overlap where the circles cross. If nothing is in both, $A \cap B = \emptyset$ .
$A^{'}$ A complement, or A dashEverything in $ξ$ that is not in $A$ . ‘Not in $A$ ’ only means something because $ξ$ fences it in: with no stated universal set there is no complement to list.

The one idea underneath all of it

Every set in a question is carved out of the same universal set $ξ$ , so the parts must add back to the whole. That single idea gives you the complement ( $n (A) + n (A^{'}) = n (ξ)$ ); it gives you the four regions of a two-set Venn diagram — only $A$ , both, only $B$ , neither — which between them account for every element exactly once; and it gives you the correction that fixes the commonest error on this topic. An element in the overlap is counted once in $n (A)$ and again in $n (B)$ , so $n (A \cup B) = n (A) + n (B) - n (A \cap B)$ , never simply $n (A) + n (B)$ .

The smallest possible example

$ξ = {1, 2, 3, 4, 5, 6}$ , with $A = {2, 4, 6}$ and $B = {4, 5, 6}$ .

$A \cap B = {4, 6}$ , $A \cup B = {2, 4, 5, 6}$ , $A^{'} = {1, 3, 5}$

The intersection keeps only what appears in both lists. The union keeps everything in either list, but writes $4$ and $6$ once, which is why $n (A \cup B) = 4$ and not $3 + 3 = 6$ . The complement is whatever is left of $ξ$ after $A$ is removed, so $A$ and $A^{'}$ between them use up all six elements. Every other question on this page is this, with longer sets or with the numbers replaced by people.

Key terms

The words a question will use

Exam wording is precise. These five carry the meaning a question depends on.

TermWhat it meansNotation
Well defined
The rule is sharp enough that everyone builds the same list. ‘The vowels of the English alphabet’ is well defined; ‘the tall students’ is not, because it is a judgement rather than a rule. This is exactly what a ‘which of these is not a set?’ question tests, and the answer is always the option that is a matter of opinion. ${a, e, i, o, u}$
Empty set
The set with no elements at all, written $\emptyset$ or ${}$ . It is not the universal set, and it is not the number zero: $n (\emptyset) = 0$ , but $\emptyset$ itself is still a set. Two sets with nothing in common have an empty intersection. $A \cap A^{'} = \emptyset$
Subset
Every element of the first set is also in the second, written $A \subset B$ . Every set is a subset of the universal set it was carved from, so $A \subset ξ$ is always true. ${2, 4} \subset {1, 2, 3, 4}$ , but ${2, 5}$ is not, because $5$ is missing from the second set. $A \subset ξ$
Number of elements
How many members a set has, written $n (A)$ . It is a count, so the answer is a plain number and not a list: $n ({4, 8, 12}) = 3$ . Read the question for which one it wants — ‘find $A \cap B$ ’ asks for the set, ‘find $n (A \cap B)$ ’ asks for the number. $n (A \cup B) = n (A) + n (B) - n (A \cap B)$
Set-builder notation
A set given by a rule instead of a list. Read the vertical bar as ‘such that’: ${x ∣ x \in ξ, x > 3}$ is ‘the values of $x$ in $ξ$ such that $x$ is greater than $3$ ’. Higher-tier questions use it to hide an inequality, so write the list out in full before answering anything. ${x ∣ x > 3}$

The method

How to work out a set language and notation

Five steps, in this order. The worked examples below point back to these numbers.

Write out the universal set $ξ$ in full, or write down its total.Everything you are allowed to use lives inside $ξ$ , and a complement cannot be listed without it. When $ξ$ is a group of people rather than a list of numbers — ‘a class of $30$ students’ — the total is the number you will be subtracting from at the end.
Turn each named set into an explicit list of members, or into a count $n (A)$ .Sets arrive as rules — ‘multiples of $3$ ’, ‘factors of $12$ ’, ${y ∣ y^{2} < 5}$ — and every mark after this depends on the list being right. Test the rule against each member of $ξ$ in turn instead of writing the list from memory.
Fill the overlap $A \cap B$ first, then subtract it from each set to get the ‘only’ regions.This is the step that stops double-counting. With $30$ students, $18$ playing football, $15$ playing rugby and $10$ playing both, the football circle holds $18 - 10 = 8$ in its ‘only’ part, not $18$ , and the rugby circle holds $15 - 10 = 5$ . Fill the middle first and every other region follows by subtraction.
Read off the region the question actually asked for.‘Only rugby’ is a single region, $5$ . ‘At least one’ is the whole union, $18 + 15 - 10 = 23$ . ‘Neither’ is the region outside both circles, which comes from the total: $30 - 23 = 7$ . Underline the deciding word — and, or, not, only, neither — before you write any number down.
Check that the parts add back to the whole.The four regions of a two-set Venn diagram must total $n (ξ)$ , and any complement must satisfy $n (A) + n (A^{'}) = n (ξ)$ . Above: $8 + 10 + 5 + 7 = 30$ . If it does not add up, the overlap was either subtracted twice or not subtracted at all, and this check takes five seconds.

Use it when

The question names a universal set — a range of numbers, a class, a survey group — and then describes one or two subsets of it, either by a rule or by a count. Listing questions (‘list the elements of $A^{'}$ ’) want the members written out inside braces; two-category survey questions (‘ $18$ play football, $15$ play rugby, $10$ play both’) want a Venn diagram with the overlap filled in first.

Do not use it when

Do not assume that every two-category problem needs overlapping circles. If the categories are disjoint, a correct Venn diagram uses separate circles or an explicitly empty intersection; if the question only asks for a fraction of a total, no set diagram may be needed at all. For example, ‘ $\frac{3}{5}$ of $180$ students are in Year $10$ , and half of those are girls’ is answered directly by $\frac{3}{5} \times 180 = 108$ before taking half. Follow the membership relationships stated in the question rather than drawing an overlap by habit.

Worked examples

6 IGCSE Maths practice questions on set language and notation, fully worked

One per objective, so the range is covered rather than the same question repeated. Every step is written out, and the answer is shown — this is study material, not a test.

Example 1

Which of the following is NOT a set?
A set is a well-defined collection of objects. [1 mark]

Understand the definition of a set

Worked solution

Step 1: A set must be 'well-defined', meaning it is clear whether an object belongs to it or not.
Step 2: Option A: The even numbers greater than 0 and less than 10 are 2, 4, 6, 8 – this is well-defined and so is a set.
Step 3: Option C: The letters in 'MATHS' are M, A, T, H, S – this is also well-defined and is a set.
Step 4: Option D: The months with 31 days can be listed unambiguously, so this is a set.
Step 5: Option B: 'Best football players' is an opinion – different people will include different names. This is NOT well-defined and therefore NOT a set.

AnswerThe set of best football players in the world

Example 2

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, write down the numbers in A ∩ B. [1 mark]

Use the set notation union, intersection, element-of and not-element-of

Worked solution

Step 1: Look at the two sets: A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.
Step 2: The symbol ∩ means 'intersection' — we need numbers that are in BOTH sets.
Step 3: Check each number in A: 1 is not in B, 2 is not in B, 3 is in B, 4 is in B.
Step 4: So the numbers that belong to both A and B are 3 and 4.
Step 5: Write them as a set: {3, 4}.

Answer{3, 4}

Example 3

The universal set $U$ contains the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Set A contains all the even numbers from $U$ . What is the complement of set A in $U$ ? [2 marks]

Understand the concept of the universal set and the empty set and the symbols for these sets

Worked solution

Step 1: The universal set $U$ contains all the numbers from 1 to 10.
Step 2: Set A is evens so A = {2, 4, 6, 8, 10}.
Step 3: The complement of A is all numbers in $U$ not in A.
Step 4: Removing {2,4,6,8,10} from {1,2,...,10} leaves {1,3,5,7,9}.

Answer{1, 3, 5, 7, 9}

Show 3 more worked examples

Example 4

The universal set is $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}$ . Set A = {multiples of 3} and set B = {factors of 12}. Find the complement of set A, represented as $A^{'}$ . [2 marks]

Understand and use the complement of a set

Worked solution

Step 1: List all members of the universal set: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
Step 2: Set A contains multiples of 3 up to 12: 3, 6, 9, 12.
Step 3: The complement of A is everything in the universal set that is NOT in A.
Step 4: Remove 3, 6, 9, 12 from the universal set.
Step 5: Remaining members: 1, 2, 4, 5, 7, 8, 10, 11.

Answer\{1, 2, 4, 5, 7, 8, 10, 11\}

Example 5

The Venn diagram shows the number of students in a class who play football (F) or rugby (R). There are 30 students in total. 12 play football, 10 play rugby, and 5 play both football and rugby. How many students play neither sport? [2 marks]

Use Venn diagrams to represent sets

Worked solution

Step 1: Add the number who play football only: 12 − $5 = 7$ .
Step 2: Add the number who play rugby only: 10 − $5 = 5$ .
Step 3: Add the number who play both: 5.
Step 4: Total who play at least one sport: $7 + 5 + 5 = 17$ .
Step 5: Students who play neither: 30 − $17 = 13$ .

Answer13

Example 6

The universal set is defined as $U = {x ∣ x \in Z, - 3 \leq x \leq 3}$ . Set $A$ is defined as $A = {y ∣ y \in U, y^{2} < 5}$ . List the elements of $A$ . Give your answer as a list of integers in increasing order separated by commas, e.g. -1, 0, 1. [3 marks]

Understand sets defined in algebraic terms, and understand and use subsets

Worked solution

Step 1: First, list all integers in the universal set: -3, -2, -1, 0, 1, 2, 3.
Step 2: For each integer, check if its square is less than 5. (-3)^ $2 = 9$ (not less than 5), (-2)^ $2 = 4$ (less than 5), (-1)^ $2 = 1$ (less than 5), 0^ $2 = 0$ (less than 5), 1^ $2 = 1$ (less than 5), 2^ $2 = 4$ (less than 5), 3^ $2 = 9$ (not less than 5).
Step 3: Collect the integers that satisfy the condition: -2, -1, 0, 1, 2.
Step 4: Write them in increasing order separated by commas: -2, -1, 0, 1, 2.

Answer-2, -1, 0, 1, 2

Exam technique

How the marks are actually awarded

Method marks are given for working a marker can follow. This is what to put on the page.

Where the marks are: Questions on this section run from $1$ to $4$ marks. One mark buys a symbol or a single set written down. Listing a complement is usually $2$ marks: one for identifying the members of the set itself, one for the accurate complement. A two-category Venn question is $3$ or $4$ marks, and the diagram is where the method marks live — a rectangle labelled $ξ$ , two overlapping circles, and the overlap filled in first will earn them even if the final subtraction goes wrong. On a ‘show that’ question the printed answer earns nothing on its own: every line of the argument has to be on the page, including the sentence saying what you are subtracting and why.
Command words to expect: List the elements ofWork outFindWrite downComplete the Venn diagramUse a Venn diagram to represent this informationShow thatExplain why
Accuracy and rounding: Nothing here rounds — sets are exact — so the marks go on form instead. Write a set inside curly brackets with commas between the elements, each element once and once only: ${2, 4, 6, 8, 10}$ . Order does not affect correctness, but writing them in increasing order makes a missing element obvious. The empty set is $\emptyset$ or ${}$ , never $0$ and never the word ‘none’. Above all, match the form to the question: $A \cap B$ asks for a set and $n (A \cap B)$ asks for a number, and a correct set written where a count was wanted still loses the accuracy mark.
Calculator: A calculator is allowed in every paper of this qualification, on both Foundation and Higher tiers, and it is close to useless here: listing, matching and counting are all done by hand. Its one honest use is the arithmetic that ends a Venn question — confirming $35 + 28 - 48 = 15$ — and it will not tell you whether $48$ should have been subtracted at all. Spend the time on the diagram instead.

Check yourself

You should now be able to:

Decide whether a described collection is a set, by testing whether its rule is well defined rather than a matter of opinion.
Read and write $\in$ and $\in /$ , and judge whether a given element belongs to $A \cup B$ or to $A \cap B$ .
Use $ξ$ for the universal set and $\emptyset$ for the empty set, and keep the two apart.
List the complement $A^{'}$ by taking every element of $ξ$ that is not in $A$ .
Draw a two-set Venn diagram inside a rectangle labelled $ξ$ , filling the overlap before the ‘only’ regions.
Read a set defined in algebraic terms, such as ${x ∣ x \in ξ, x^{2} < 5}$ , and list its elements.
Decide whether one set is a subset of another, and write it as $A \subset B$ .
Fill a Venn diagram with the number of elements in each region, and work backwards from a total to a missing overlap.
Use $n (A)$ for a count, and apply $n (A \cup B) = n (A) + n (B) - n (A \cap B)$ instead of adding the two totals.
Answer ‘at least one’, ‘both’, ‘only’ and ‘neither’ questions from a two-category survey.

Specification coverage

The 9 things 4MA1 asks you to do

The assessable objectives for numbers and the number system — set language and notation, in the order the specification lists them.

Understand the definition of a set
Use the set notation union, intersection, element-of and not-element-of
Understand the concept of the universal set and the empty set and the symbols for these sets
Understand and use the complement of a set
Use Venn diagrams to represent sets
Understand sets defined in algebraic terms, and understand and use subsets

Show 3 more objectives

Use Venn diagrams to represent sets and the number of elements in sets
Use the notation n(A) for the number of elements in the set A
Use sets in practical situations

See who is stuck on set language and notation before you teach it.

Create a class, share the 8-character code, and set set language and notation as practice. Your students work through all 32 approved practice questions — not just the 6 on this page — and you see which objectives the class has not got yet. Free for individual students and teachers.

How Learnly works for teachers

Set up your class — free Practise on your own

What you need to be able to do already

List the members of a set from a rule — multiples, factors, primes, and the integers between two limits

Read an inequality as the numbers it allowsInequalities

Write one part of a total as a fraction, and find a fraction of a totalFractions

What is set language and notation?

The notation, and how to read it

The words a question will use

Well defined

Empty set

Subset

Number of elements

Set-builder notation

How to work out a set language and notation

6 IGCSE Maths practice questions on set language and notation, fully worked

Which of the following is NOT a set? A set is a well-defined collection of objects. [1 mark]

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, write down the numbers in A ∩ B. [1 mark]

The universal set U contains the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Set A contains all the even numbers from U. What is the complement of set A in U? [2 marks]

The universal set is U={1,2,3,4,5,6,7,8,9,10,11,12}. Set A = {multiples of 3} and set B = {factors of 12}. Find the complement of set A, represented as A′. [2 marks]

The Venn diagram shows the number of students in a class who play football (F) or rugby (R). There are 30 students in total. 12 play football, 10 play rugby, and 5 play both football and rugby. How many students play neither sport? [2 marks]

The universal set is defined as U={x∣x∈Z,−3≤x≤3}. Set A is defined as A={y∣y∈U,y2<5}. List the elements of A. Give your answer as a list of integers in increasing order separated by commas, e.g. -1, 0, 1. [3 marks]

How the marks are actually awarded

You should now be able to:

The 9 things 4MA1 asks you to do

See who is stuck on set language and notation before you teach it.

Which of the following is NOT a set?
A set is a well-defined collection of objects. [1 mark]

The universal set $U$ contains the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Set A contains all the even numbers from $U$ . What is the complement of set A in $U$ ? [2 marks]

The universal set is $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}$ . Set A = {multiples of 3} and set B = {factors of 12}. Find the complement of set A, represented as $A^{'}$ . [2 marks]

The universal set is defined as $U = {x ∣ x \in Z, - 3 \leq x \leq 3}$ . Set $A$ is defined as $A = {y ∣ y \in U, y^{2} < 5}$ . List the elements of $A$ . Give your answer as a list of integers in increasing order separated by commas, e.g. -1, 0, 1. [3 marks]