Set Notation

Read and write set symbols: list the members of a set, use the symbols for member, empty set and subset, build a set from clues and count with n( ).

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 78 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What Set Notation Means

A set is a list of things inside curly brackets, like ${2, 4, 6}$ . Each thing in it is a member. Set notation is a short way to talk about sets.

Symbol	Means
$E$	the universal set: everything in the question
$x \in A$	x is a member of A
$x \in / A$	x is not a member of A
$A \cap B$	in A and in B (the overlap)
$A \cup B$	in A or B or both
$A^{'}$	not in A (the complement)
$C \subseteq A$	C is a subset of A: every member of C is in A (Extended/Higher; Edexcel writes ⊂)
$C ⊈ A$	C is not a subset of A: at least one member of C is not in A (Extended/Higher)
$\emptyset$	the empty set: no members
$n (A)$	how many members A has

Memory trick

Cap and cup: $\cap$ looks like a bridge joining the two sets (and). $\cup$ is a cup that holds everything from both (or).

Listing the Members of a Set

The most common question, 1 or 2 marks: you get $E$ , A and B as rules. Write each set out in full first, then read off the one they ask for.

Worked example

1
$E = {10, 11, \dots, 20}$ — Write out the universal set. Every answer must come from these numbers.
2
$A = {12, 15, 18}$ — A is the multiples of 3. Go through 10 to 20 and keep the ones in the 3 times table.
3
$B = {11, 13, 15, 17, 19}$ — B is the odd numbers. Keep every number from 10 to 20 that does not divide by 2.
4
$A \cap B = {15}$ — $\cap$ means in both lists. Only 15 is in A and in B.
5
$A \cup B = {11, 12, 13, 15,$ — $\cup$ means in A or B or both. Go through 10 to 20 in order and keep every number in either list.
6
$17, 18, 19}$ — 15 is in both lists, but you write it only once.
7
$A^{'} \cap B = {11, 13, 17, 19}$ — $A^{'}$ means not in A. So start with B and remove 15, because 15 is in A.
8
$(A \cup B)^{'} = {10, 14, 16, 20}$ — Outside A and outside B: in neither set. These are the even numbers that are not multiples of 3.

Common mistake

Writing 15 twice in $A \cup B$ . A number in both sets is listed once.

Key idea

A dash means not. $A^{'}$ is everything in $E$ that is not in A, so it includes numbers in neither set too.

Sets Written as {x : …}

Exams often give a set as a rule in curly brackets, like ${x : x is a factor of 12}$ . Read the colon as “such that”. Always turn the rule into a list before you answer.

Worked example

1
$E = {x : x is an integer, 1 \leq x \leq 12}$ — Read it in words: all the whole numbers x such that x is from 1 to 12.
2
$E = {1, 2, 3, \dots, 12}$ — The sign ≤ means “or equal to”, so 1 and 12 are both included.
3
$A = {x : x is a factor of 12}$ — A keeps only the numbers in $E$ that divide 12 exactly.
4
$A = {1, 2, 3, 4, 6, 12}$ — Test each number: 12 ÷ 4 = 3 is whole, so 4 is in. 12 ÷ 5 is not whole, so 5 is out.
5
$B = {x : 5 < x \leq 9}$ — Two different signs. < means 5 is not included, ≤ means 9 is included.
6
$B = {6, 7, 8, 9}$ — Start at the first whole number above 5, and stop at 9.
7
$A \cap B = {6}$ — Now it is an ordinary question. Only 6 is in both lists.

Common mistake

Putting 5 in B. The sign < means “more than”, so 5 itself is left out. Only ≤ and ≥ include the end number.

Member, Empty Set and Subset

The symbol question: write the right symbol in the gap, or say if a statement is true and give a reason. Use the sets from Listing the Members: the numbers 10 to 20, A = multiples of 3 and B = odd numbers.

Worked example

1
$18 \in A$ — 18 = 3 × 6, so it is a multiple of 3. That makes it a member of A.
2
$14 \in / B$ — 14 is even, so it is not a member of B, the odd numbers.
3
$C = {10, 20}$ — A new set to test: just the numbers 10 and 20.
4
$A \cap C = \emptyset$ — Neither 10 nor 20 is a multiple of 3, so no number is in both. The answer is the empty set.
5
${12, 18} \subseteq A$ — 12 and 18 are both in A, so this small set is a subset of A.

Rule

To show a statement is wrong, name one member that breaks it. “ $A \cap B = \emptyset$ ” is false because 15 is in A and in B. To show a set really is empty, say no number is in all of the sets at once.

Common mistake

Writing $A \cap C = 0$ . The answer is a set, so write $\emptyset$ , not the number 0.

Key idea

Why does the empty set never feel left out? Because $\emptyset$ is a subset of every set. It has no members, but it is invited everywhere!

Building a Set From Clues

The 2-mark question: find a set C that fits some clues. Turn each clue into words, then choose numbers that pass every clue.

Worked example

1
$C \subseteq B$ — Clue 1: C is a subset of B. So every member of C must be odd.
2
$A \cap C = \emptyset$ — Clue 2: C shares nothing with A. So C has no multiples of 3.
3
$A^{'} \cap B = {11, 13, 17, 19}$ — The numbers that pass both clues: odd (in B) and not in A. 15 is removed because it is a multiple of 3.
4
$n (C) = 3$ — Clue 3: C has exactly 3 members.
5
$C = {11, 13, 17}$ — Pick any 3 of the four allowed numbers. Any choice like this scores full marks.

Common mistake

Putting 15 in C. It is odd, but it is also in A, so it breaks the second clue.

Counting With n( )

The 1-mark question: find how many members a set has. You give one number, you do not list them.

Worked example

1
$n (E) = 40, n (A) = 25$ — Read the two totals from the question: 40 in the box, 25 in A.
2
$n (A^{'}) = n (E) - n (A)$ — Everything not in A is the rest of the box, so take A away from the total.
3
$n (A^{'}) = 40 - 25 = 15$ — 40 − 25 = 15 members are not in A.

From a Venn diagram, add the numbers in the regions the set covers. Say it shows 12 in A only, 13 in both, 9 in B only and 6 outside.

Worked example

1
$n (A^{'} \cap B) = 9$ — In B and not in A. That is only the B only region, so 9.
2
$n (A \cap B^{'}) = 12$ — In A and not in B. That is only the A only region, so 12.
3
$n ((A \cup B)^{'}) = 6$ — Not in A and not in B: the part outside both circles, so 6.

Common mistake

Listing the members when the question says $n (\dots)$ . It wants one number: how many.

The Union Formula

When a question gives totals but no list, this formula links the union and the overlap:

Formula

n (A \cup B) = n (A) + n (B) - n (A \cap B)

$n (A)$ and $n (B)$ are the totals of each set. $n (A \cap B)$ is the number in both. You take it away because adding the two totals counts the overlap twice.

Worked example

1
$n (E) = 35, n (A) = 22, n (M) = 18, n (A \cap M) = 9$ — A class of 35: 22 do art, 18 do music, 9 do both.
2
$n (A \cup M) = 22 + 18 - 9$ — Add the two totals, then take away the 9 that were counted twice.
3
$n (A \cup M) = 31$ — 22 + 18 = 40, and 40 − 9 = 31 do art or music or both.
4
$n ((A \cup M)^{'}) = 35 - 31 = 4$ — The rest of the class do neither, so take 31 from 35.

Harder: “30 people, 21 like tea, 16 like coffee. What is the smallest possible number who like both?”

Worked example

1
$21 + 16 = 37$ — Add the totals. 37 is more than 30, so some people must be counted twice.
2
$n (T \cup C) \leq 30$ — Nobody is outside the group of 30, so the union is at most 30.
3
$n (T \cap C) = 37 - n (T \cup C)$ — Rearrange the formula to get the overlap on its own.
4
$n (T \cap C) \geq 37 - 30 = 7$ — The overlap is smallest when the union is as big as it can be, 30. So at least 7 like both.

Key idea

The largest possible number in neither: put the smaller set inside the bigger one. Then 21 like tea (some also coffee) and 30 − 21 = 9 like neither.

Naming a Shaded Region

The 1-mark question: “Use set notation to describe the shaded region.” Ask two things: which circle is it inside, and which is it outside?

Shaded part	Write
the overlap	$A \cap B$
both whole circles	$A \cup B$
outside both circles	$(A \cup B)^{'}$
everything except the overlap	$(A \cap B)^{'}$
A only	$A \cap B^{'}$
everything outside circle B	$B^{'}$

Worked example

1
$shaded: A only$ — Look first. The shading is inside circle A, but not in the overlap.
2
$inside A \to A$ — Inside a circle means in that set.
3
$outside B \to B^{'}$ — Outside a circle means not in it, so use the dash.
4
$A \cap B^{'}$ — Both must be true at once, so join them with $\cap$ (and).

Common mistake

Writing “A only” or “A − B”. Use only the set symbols $\cap$ , $\cup$ and the dash.

Key idea

Check your answer region by region. Each of the four parts of the diagram must be shaded if, and only if, your notation includes it.

Which Method? Read the Question

Look at the symbol and what the question asks for.

You see	You do
${x : \dots}$	turn the rule into a list first
List the members of	write A and B in full, then read off
$A \cap B$	numbers in both lists
$A \cup B$	numbers in either list, once each
$A^{'}$	everything in the box that is not in A
Describe the shaded region	inside a circle: the letter; outside: the dash
Write a symbol in the gap	member, not a member, or empty set
Find a set C with clues	turn each clue into words
$n (\dots)$	count or add, one number
Totals and the number in both	$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
Fill a Venn diagram	Venn Diagrams
An unknown x in the regions	Set Problems
Three circles	Three-Set Venn Diagrams

Key idea

Read the symbol out loud in words first: and, or, not. Then the question is easy.

Practise set notation now

Interactive visual aids, then real past-paper questions marked on your working like the exam. Free to start.

Practise free on Lemmara

Real past-paper questions on set notation

Each with a full worked solution.

All 78 questions on this topic →

Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education or Pearson Edexcel. The lessons and worked examples are Lemmara’s own.