Venn Diagrams

Draw a two-set Venn diagram from lists or counts, then read off the members or the number in any set.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 99 real past-paper questions below

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What a Venn Diagram Shows

A Venn diagram is a box with two circles that overlap. The box holds everything, called $E$ . Each circle is one set.

Region	Holds
Left circle only	in A, not in B
Overlap	in both, $A \cap B$
Right circle only	in B, not in A
Outside both circles	in neither

Rule

Every member of $E$ goes in exactly one region. Never write a number twice.

Key idea

Two special drawings. If every member of C is also in A, written $C \subset A$ , draw circle C inside circle A. If two sets share nothing, written $A \cap B = \emptyset$ , draw the circles apart, not overlapping.

Key idea

The English logician John Venn published these overlapping circles in 1880. He never called them Venn diagrams: he called them Eulerian circles, after the older mathematician Euler. The name "Venn diagram" only appeared in 1918.

Filling a Venn Diagram From Lists

The 3-mark question: you get $E$ , A and B as lists or rules. Fill the overlap first, then the rest of each circle, then the outside.

Worked example

1
$E = {1, 2, \dots, 12}$ — The box. Every number from 1 to 12 must end up in exactly one region.
2
$A = {1, 2, 3, 4, 6, 12}$ — Write A out in full. The factors of 12 are the numbers that divide 12 exactly.
3
$B = {1, 3, 5, 7, 9, 11}$ — Write B out in full: the odd numbers from 1 to 12.
4
$A \cap B = {1, 3}$ — Look for numbers in both lists. 1 and 3 are in both, so they go in the overlap first.
5
$A only = {2, 4, 6, 12}$ — Take 1 and 3 out of A. The numbers left are in A but not in B.
6
$B only = {5, 7, 9, 11}$ — Take 1 and 3 out of B. The numbers left are in B but not in A.
7
$outside = {8, 10}$ — Cross off every number used so far. 8 and 10 are in neither list, so they go inside the box but outside both circles.
8
$4 + 2 + 4 + 2 = 12$ — Check: A only, overlap, B only and outside hold 12 numbers, the same as the box. None is missing or written twice.

Common mistake

Writing 1 and 3 in the overlap and again in A only. Each number goes in one place. And do not forget 8 and 10 outside the circles.

Filling From Set Notation

Some questions give the sets in symbols, like $A \cap B$ and $A^{'}$ , instead of rules. Turn each symbol into words, then fill the regions it points to.

Say $E = {1, 2, \dots, 10}$ , $A \cap B = {4, 8}$ , $A^{'} = {1, 3, 5, 7, 9, 10}$ and $B^{'} = {1, 2, 6, 7, 10}$ .

Worked example

1
$A \cap B = {4, 8}$ — These are in A and in B, so 4 and 8 go straight in the overlap.
2
$A^{'} = {1, 3, 5, 7, 9, 10}$ — These are not in A. So each one is either in B only or outside both circles.
3
$B^{'} = {1, 2, 6, 7, 10}$ — These are not in B. So each one is either in A only or outside both circles.
4
$A^{'} \cap B^{'} = {1, 7, 10}$ — 1, 7 and 10 are in both lists, so they are in neither set. They go outside the circles.
5
$B only = {3, 5, 9}$ — The rest of A′ is not in A and not outside, so 3, 5 and 9 must be in B only.
6
$A only = {2, 6}$ — The rest of B′ is not in B and not outside, so 2 and 6 must be in A only.
7
$2 + 2 + 3 + 3 = 10$ — Check: A only, overlap, B only and outside use all 10 numbers in the box, each once.

Common mistake

Putting every number of A′ into circle B. A′ also holds the numbers outside both circles, so find those first.

Key idea

If you are given $A \cup B$ instead, the numbers of $E$ missing from it are the ones outside both circles.

Reading Members From a Venn Diagram

The 1-mark question: list the members of a set. Work out which regions it covers, then read the numbers in them. These use the diagram from Filling a Venn Diagram From Lists: A is the factors of 12, B is the odd numbers, and the box is 1 to 12.

Steps

1
$A \cup B$ : both circles: 1, 2, 3, 4, 5, 6, 7, 9, 11, 12
2
$(A \cup B)^{'}$ : outside both: 8, 10
3
$B^{'}$ : A only and outside: 2, 4, 6, 8, 10, 12
4
$A \cap B^{'}$ : A only: 2, 4, 6, 12

Common mistake

For $B^{'}$ you need everything not in B. That includes the numbers outside both circles, not just A only.

Key idea

"Explain why $A \cap C = \emptyset$ ": say no number is in both A and C, so the circles share nothing.

Shading a Region

The 1-mark shading question. A two-circle diagram has four regions: A only, the overlap, B only and outside. Test each one: does it fit the symbols? Shade only the yes regions.

Memory trick

$\cap$ means and, $\cup$ means or, and a dash ′ means not.

1. Shade $(A \cap B)^{'}$

Worked example

1
$A \cap B = overlap$ — Do the bracket first. In A and in B is the overlap.
2
$(A \cap B)^{'} = not the overlap$ — The dash outside the bracket means not. So you want every region except the overlap.
3
$shade A only, B only, outside$ — Three regions fit. Shade them and leave only the overlap white.

2. Shade $A^{'} \cup B$

Worked example

1
$A^{'} = B only + outside$ — A′ means not in A. That is B only and the part outside both circles.
2
$B = B only + overlap$ — B is the whole of circle B: B only and the overlap.
3
$A^{'} \cup B = either list$ — ∪ means or, so a region fits if it is in either list. Count each region once.
4
$shade overlap, B only, outside$ — Only A only is left white, because it is in A and not in B.

Common mistake

Shading only where the two lists share regions. That is ∩ (and). For ∪ (or), shade every region in either list.

Key idea

The same yes-or-no test works with three circles. There are 8 regions to test instead of 4.

Naming a Shaded Region

The reverse question: a region is already shaded, and you write it in symbols. Say it in words first, then turn each word into a symbol.

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

Worked example

1
$shaded: A only$ — The question. The shaded part is inside circle A but does not touch circle B.
2
$in A$ — It is inside circle A, so one fact is in A.
3
$not in B \to B^{'}$ — It is outside circle B, so the other fact is not in B, written B′.
4
$A \cap B^{'}$ — Both facts are true at once. And is ∩, so the answer is A ∩ B′.

Common mistake

Writing $(A \cap B)^{'}$ for the part outside both circles. $(A \cap B)^{'}$ is everything except the overlap. Outside both is $(A \cup B)^{'}$ .

Key idea

“Describe in words” works the same way, backwards. If B is the even numbers and C the multiples of 5, then $B^{'} \cap C$ is not even and a multiple of 5: the odd multiples of 5.

Numbers in the Regions

The 3-mark counting question: you get totals, not lists. Put the number in both in the middle first, then take it away from each total.

Worked example

1
$40 students: 23 swim, 18 run, 9 both$ — The question. The 23 and the 18 both include the 9 who do both sports.
2
$both = 9$ — Write 9 in the overlap first, because it is part of both totals.
3
$swim only = 23 - 9 = 14$ — Take the 9 who also run away from the 23 swimmers. 14 swim but do not run.
4
$run only = 18 - 9 = 9$ — Do the same for running: 18 runners take away the 9 in the overlap leaves 9.
5
$14 + 9 + 9 = 32$ — Add the three regions inside the circles. 32 students do at least one sport.
6
$neither = 40 - 32 = 8$ — Everyone else is in neither circle, so 8 go outside.

Now any $n (\dots)$ question is adding regions. For example, $n (S^{'}) = 9 + 8 = 17$ and $n (S \cup R) = 14 + 9 + 9 = 32$ .

Common mistake

Writing 23 in swim only. The 23 includes the 9 who do both, so swim only is 14.

Key idea

Write each subtraction next to the diagram. If one region ends up wrong, your working can still earn a mark. No region can be negative: if you get one, check which total includes the overlap.

Shortcut: The n(A ∪ B) Formula

Textbooks print one formula that links the totals. It is the quickest route when the question gives $n (A \cup B)$ or the number in neither.

Formula

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

$n (A)$ is how many are in A, $n (B)$ how many are in B, $n (A \cap B)$ how many are in both, and $n (A \cup B)$ how many are in A or B or both. You take the overlap away once because adding n(A) and n(B) counts it twice.

Worked example

1
$n (E) = 50, n (A) = 28, n (B) = 19$ — The question, and 9 are in neither set. We need the overlap before we can fill anything.
2
$n (A \cup B) = 50 - 9 = 41$ — Everyone who is not outside the circles is in A or B or both, so 41.
3
$41 = 28 + 19 - n (A \cap B)$ — Put the three numbers you know into the formula.
4
$41 = 47 - n (A \cap B)$ — Add the two totals first: 28 + 19 = 47.
5
$n (A \cap B) = 47 - 41 = 6$ — 47 is 6 more than 41, because 6 people were counted twice. They are in both.
6
$A only = 28 - 6 = 22$ — Take the overlap away from A's total.
7
$B only = 19 - 6 = 13$ — Take the overlap away from B's total.
8
$22 + 6 + 13 + 9 = 50$ — Check: the four regions add up to the 50 in the box.

Common mistake

Putting 50 into the formula as $n (A \cup B)$ . The 9 outside the circles are not in A or B, so $n (A \cup B)$ is 41.

Which Method? Read the Question

Look at what the question gives you and what it wants.

You see	You do
Lists or rules for A and B	overlap first, then each circle, then outside
Totals and a number in both	overlap first, take it from each total
List the members of a set	find its regions, read the numbers
Find n( ) from a diagram	add the numbers in its regions
Sets given in symbols, like $A^{'} = {\dots}$	overlap first, then what each list leaves out
Shade a set, like $(A \cap B)^{'}$	test each region: yes or no
Name a shaded region	say it in words, then in symbols
$n (A \cup B)$ or the number in neither	$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
An unknown x in the regions	Set Problems
Three circles	Three-Set Venn Diagrams
What does a symbol mean?	Set Notation

Key idea

Check at the end: the numbers in all four regions must add up to the total in the box.

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