Sets

A checkpoint was used to test a total of 20 trucks. 6 trucks did not pass the brakes test ($B$). 7 trucks did not pass the lights test ($L$). 9 trucks passed both the brakes and lights tests. A Venn diagram is displayed with two overlapping circles labelled $B$ and $L$ inside a universal set.

Feb/March 2017

The universal set $\xi$ contains a Venn diagram for the sets $X$, $Y$ and $Z$.

Feb/March 2018

The universal set is $\mathcal{C} = \{1,2,3_toggle,4,5,6,7,8,9,10,11,12,13,14\}$. Set $F$ is defined by $F = \{x : x$ is a factor of 14$\}$. Set $P = \{x : x$ is a prime number less than 14$\}$.

Feb/March 2020

In this year, each of the 40 students has travelled by at least one of plane ($P$), train ($T$) or boat ($B$). 7 travelled by plane only. 11 travelled by train only. 9 travelled by boat only. $n(P \cap T) = 8$, $n(B \cap T) = 3$, $n(B \cap P) = 6$.

Feb/March 2020

Complete the clock diagram so that it shows the time when the cakes are ready.

Feb/March 2021

Inside the universal set, a Venn diagram displays two sets named $A$ and $B$.

Feb/March 2023

A Venn diagram displays two overlapping sets, A and B, within the universal set.

Feb/March 2023

$\xi = \{\text{students in a class}\},\; P = \{\text{students who study Physics}\},\; C = \{\text{students who study Chemistry}\}$ Also, $n(\xi)=24,\; n(P)=17,\; n(C)=14,\; n(P\cap C)=9$

Feb/March 2023

$x$ is an integer. $\mathcal{E} = \{x : 1 \le x \le 10\}$, $P = \{x : x$ is an even number$\}$, $Q = \{x : x$ is a multiple of $5\}$. The Venn diagram below shows sets $P$ and $Q$ within the universal set $\mathcal{E}$.

Feb/March 2024

A university mathematics department employs 120 people to teach. Some details are displayed in the table. One fifth of the people are professors. 30% of the people work part-time.

Feb/March 2024

A Venn diagram displays three intersecting sets $A$, $B$ and $C$ within the universal set $\xi$.

Feb/March 2025

Let $\xi = \{x: 1 \le x \le 12,\ x\text{ is an integer}\}$, with $M$ as the odd numbers and $N$ as the multiples of 3. The Venn diagram below places sets $M$ and $N$ inside $\xi$.

May/June 2015

A Venn diagram is displayed. In a class of 30 students, 25 take French (F) and 18 take Spanish (S). One student takes neither French nor Spanish.

May/June 2015

30 students were surveyed about whether they owned a bicycle (B), a mobile phone (M) and a computer (C). Their responses are displayed in the Venn diagram. The diagram gives these counts: In B only: 2; B∩M: 4; M only: x; B∩C: 1; B∩M∩C: 7; M∩C: 6; C only: 3; outside all sets: 2.

May/June 2015

The Venn diagram indicates how many elements fall in each section. Set $A$ only has 3 elements, $A \cap B$ has 7 elements, $B$ only has 12 elements, and the part outside both sets has 5 elements.

May/June 2016

$\mathcal{E}=\{x:2\le x\le16,\ x\text{ is an integer}\}$ $M=\{\text{even numbers}\}$ $P=\{\text{prime numbers}\}$

May/June 2016

$\mathcal{E} = \{ \text{students in a class} \}$, $P = \{ \text{students who study physics} \}$, $C = \{ \text{students who study chemistry} \}$. The Venn diagram gives the student counts. In $P$ only: $5$. In $P \cap C$: $11$. In $C$ only: $8$. Outside both circles: $7$.

May/June 2017

$\mathcal{E} = \{21, 22, 23, 24, 25, 26, 27, 28, 29, 30\}$, $A = \{x : x \text{ is divisible by } 3\}$, $B = \{x : x \text{ is prime}\}$, $C = \{x : x \le 25\}$.

May/June 2017

The Venn diagram contains a universal set $\xi$ and two intersecting circles named $P$ and $Q$.

May/June 2018

Calculate the fee for 2016.

May/June 2018

The Venn diagrams are displayed. For part (a), a rectangle is used to show the universal set, with two overlapping circles marked $X$ and $Y$. For part (b), the Venn diagram contains these values: In $M$ only: 3. In $M \cap P$: 6. In $P$ only: 12. In $M \cap C$: 5. In $M \cap P \cap C$: 2. In $P \cap C$: 10. In $C$ only: 7. Outside all circles: 23.

May/June 2019

Forty children were surveyed about whether they have a computer, a phone, or both. The Venn diagram gives the results. The regions are: computer only $7$, both $8$, phone only $23$, neither $2$.

May/June 2019

Let $\xi = \{\text{students in a school}\}$. $F = \{\text{students who play football}\}$. $B = \{\text{students who play baseball}\}$. Altogether, the school contains 240 students. 120 of the students play football. 40 of the students play baseball. 90 play football but not baseball.

May/June 2019

A cohort of 120 students sit two tests, mathematics ($M$) and English ($E$). The following information is given about how many students pass mathematics and how many pass English: • 61 students pass mathematics. • 27 students pass both mathematics and English. • 19 students do not pass mathematics and do not pass English. A Venn diagram is displayed with two overlapping circles labelled $M$ and $E$ inside a rectangle.

May/June 2021

$\mathcal{E} = \{\text{integers exceeding }2\}$ $A = \{\text{primes}\}$ $B = \{\text{odd integers}\}$ $C = \{\text{perfect squares}\}$

May/June 2021

$\mathcal{E} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$ $E = \{x : x \text{ is an even number}\}$ $M = \{x : x \text{ is a multiple of } 3\}$ The Venn diagram below displays sets $E$ and $M$ inside $\mathcal{E}$.

May/June 2021

Shade the region $P' \cup Q$ in the Venn diagram.

May/June 2021

A class of $40$ students is represented in a Venn diagram for physics ($P$), mathematics ($M$) and geography ($G$). The diagram contains $4$ in $P$ only, $11$ in $M$ only, $6$ in $G$ only, $2$ in $P \cap M$ only, $3$ in $P \cap G$ only, $5$ in $M \cap G$ only, and $9$ in $P \cap M \cap G$. The shaded area contains everything in $M$ and everything in $G$, apart from the region that belongs only to $P$.

May/June 2022

A Venn diagram displays a universal set containing two overlapping circles labelled A and B. In A only, the number is 25. In the intersection, the number is 33. In B only, the number is 12. The region outside both circles, yet still within the universal set, is 3.

May/June 2022

State the shaded region using set notation.

May/June 2022

$\xi = \{a, b, c, d, e, f, g, h, i, j, k\}$ $F = \{a, c, e, f\}$ $B = \{a, b, c, k\}$

May/June 2023

$\mathbb{Z} = \{x : 1 \leq x < 20\}$, $E = \{\text{even numbers}\}$, $M = \{\text{multiples of }5\}$.

May/June 2023

A Venn diagram presents universal set $\mathscr{E}$, together with three intersecting circles marked $A$, $B$ and $C$.

May/June 2023

$\mathscr{E}=\{x:1\le x\le 20\}$ $E=\{\text{even numbers}\}$ $M=\{\text{multiples of }5\}$

May/June 2023

For part (a), $\mathcal{E} = \{a, b, e, g, l, m, o, r, t, y\}$, $P = \{a, b, e, g, l, r\}$ and $Q = \{e, g, m, o, r, t, y\}$. The diagram shows two intersecting circles marked $P$ and $Q$ within a rectangle marked $\mathcal{E}$. For part (b), the figure shows two intersecting circles marked $A$ and $B$ inside a rectangle labelled $\mathcal{E}$.

May/June 2023

List the members of $X$.

May/June 2023

The Venn diagram shows a universal set that contains two overlapping circles labelled A and B.

May/June 2024

$\mathcal{E}=\{x:x\text{ is a natural number less than }12\}$. $S=\{1,4,7,10\}$. $T=\{1,3,5,7,9,11\}$. The Venn diagram has circles labelled S and T. The number 10 appears in S only, while 2 is placed outside both circles, and 8 is also placed outside both circles.

May/June 2024

Complete the Venn diagram shown.

May/June 2024

In the Venn diagram, shade the area $M' \cap N'$.

May/June 2024

$W = \{$students who walk to school$\}$ $G = \{$students who wear glasses$\}$ There are 20 students in the class. 8 of them walk to school. 3 both wear glasses and walk to school. 2 neither wear glasses nor walk to school.

May/June 2024

The Venn diagram gives data on how many students are in a class. Some take English (E), some take French (F), some take Spanish (S), and some take none of these languages.

May/June 2024

A Venn diagram contains two overlapping circles, A and B, placed within the universal set.

May/June 2024

Part (a) presents a Venn diagram of two intersecting sets, $A$ and $B$, within the universal set $\mathcal{E}$. Part (b) presents a Venn diagram of three intersecting sets, $P$, $Q$ and $R$, where the area belonging to set $R$ is shaded.

May/June 2024

The universal set $\mathcal{E}$ is displayed. Sets J and K overlap, and the shaded part covers every point in circle J or circle K, including the common region.

May/June 2025

A music group contains 30 members. 7 members play a flute (F) and play a clarinet (C). 12 members play a flute. 1 member does not play a flute and does not play a clarinet.

May/June 2025

Li asks 28 students whether they speak English ($E$) and whether they speak Spanish ($S$). 15 students speak English. 12 students speak Spanish. 6 students speak neither English nor Spanish. A Venn diagram is displayed with two overlapping circles marked $E$ and $S$ inside a universal set.

May/June 2025

The universal set $\mathcal{E}$ contains the sets $A$, $B$ and $C$ in the Venn diagram.

May/June 2025

Let $\mathcal{E} = \{x : x$ is an integer and $1 \le x \le 12\}$. Define $E = \{\text{even numbers}\}$ and $M = \{\text{multiples of }3\}$.

May/June 2025

The Venn diagram presents a universal set containing two overlapping circles labelled $A$ and $B$.

Oct/Nov 2015

The Venn diagram indicates how many students take French ($F$), Spanish ($S$) and Arabic ($A$). The regions show: $7$ in $F$ only, $5$ in $S$ only, $8$ in $A$ only, $4$ in $F \cap S$, $2$ in $F \cap A$, $3$ in $S \cap A$, $1$ in $F \cap S \cap A$, and $0$ outside all sets.

Oct/Nov 2015

The Venn diagram gives the number of elements in each set. In set $P$, there are 3 in the part only in the left circle, the overlap contains 5, set $Q$ has 10 in the part only in the right circle, and 9 lie outside both sets but still within the universal set.

Oct/Nov 2015

$n(A) = 7$, $n(B) = 6$, $n(\mathcal{E}) = 10$, $n(A \cup B)' = 1$.

Oct/Nov 2016

The Venn diagram indicates how many people enjoy films ($F$), music ($M$) and reading ($R$). The regions show these counts: $8$ in $F$ only, $4$ in $M$ only, $2$ in $R$ only, $2$ in $F\cap M$, $1$ in $F\cap R$, $0$ in $M\cap R$, $3$ in $F\cap M\cap R$, and $2$ outside all three sets.

Oct/Nov 2016

$A$ = {integers} $B$ = {irrational numbers} $\mathcal{E} = \{7,\; 9.3,\; \pi,\; \frac{5}{9},\; 2\sqrt{8}\}$.

Oct/Nov 2016

Write down a set $P$ for which $P \subset Q$.

Oct/Nov 2017

$C = \{x : x \text{ is an integer and } 5 \le x \le 12\}$ together with $D = \{5, 10\}$.

Oct/Nov 2018

The Venn diagram gives data on the number of students who take Music (M), Drama (D) and Geography (G). The values shown are 5 in M only, 8 in D only, 12 in G only, 3 in $M \cap D$ only, 4 in $M \cap G$ only, 9 in $D \cap G$ only, and 2 in all three.

Oct/Nov 2018

The universal set is $\xi = \{0, 1, 2, 3, 4, 5, 6\}$, while $A = \{0, 2, 4, 5, 6\}$ and $B = \{1, 2, 5\}$.

Oct/Nov 2019

The Venn diagram depicts a universal set $\xi$ that contains two overlapping circles labelled $A$ and $B$.

Oct/Nov 2019

(a) $M = \{ x : x$ is an integer with $2 \le x < 6 \}$

Oct/Nov 2019

$\xi = \{\text{children in a group}\}$ $R = \{\text{children who own a rabbit}\}$ $H = \{\text{children who own a hamster}\}$ The group contains $40$ children. $19$ of the children own a rabbit. $27$ of the children own a hamster.

Oct/Nov 2020

The Venn diagram contains three intersecting circles, $M$, $N$ and $P$, set inside a universal set.

Oct/Nov 2020

The diagram presents a universal set as a rectangle with two intersecting circles, with $A$ on the left and $B$ on the right. The shaded area is the section of circle $B$ that lies outside the overlap with circle $A$.

Oct/Nov 2020

Write down one factor of 54.

Oct/Nov 2020

Commonwealth

Oct/Nov 2020

A village has 50 families. $C = \{\text{families who own a car}\}$. $B = \{\text{families who own a bicycle}\}$. 23 families own a car. 10 families own both a car and a bicycle. 6 families own neither a car nor a bicycle.

Oct/Nov 2021

Complete the Venn diagram by filling in the missing values.

Oct/Nov 2021

There are two Venn diagrams. In the first, the sets $A$ and $B$ are shown, and the highlighted region is $A' \cup B'$. In the second, the sets $C$, $D$ and $E$ are shown, and the highlighted region is $(C \cup D) \cap E'$.

Oct/Nov 2021

A Venn diagram is displayed with universal set $\xi$. Within set $P$ are b and c. Set $Q$ includes e, f and g. The overlap of $P$ and $Q$ contains d. The region inside $\xi$ but outside both circles contains a.

Oct/Nov 2021

The Venn diagram illustrates the elements of the sets $\mathcal{E}$, $P$ and $Q$. The items in $P$ only are $a$, $b$, $c$. The common region contains $d$. The items in $Q$ only are $e$ and $f$. The element lying outside both sets in $\mathcal{E}$ is $g$.

Oct/Nov 2022

$\xi = \{\text{people in a group}\}$ $B = \{\text{people who own a bicycle}\}$ $C = \{\text{people who own a car}\}$ A group contains 120 people. 21 people own a bicycle. 15 people own both a bicycle and a car. 35 people own neither a bicycle nor a car.

Oct/Nov 2022

The diagrams illustrate Venn diagrams. In (a), the two sets $A$ and $B$ overlap within the universal set. In (b), the elements are arranged as follows: In $X$ only: $a, h, f$ In $X \cap Y$: $c, s, d$ In $Y$ only: $p, b, g$ Outside both sets: $m, e$.

Oct/Nov 2022

The Venn diagram displays the members of the sets $\xi$, $P$ and $Q$. The items shown are: in $P$ only: $a, b, c$; in $P \cap Q$: $d$; in $Q$ only: $e, f$; and, outside both circles but still inside $\xi$: $g$.

Oct/Nov 2022

In each Venn diagram, shade the region specified by $G \cap H'$ and $(J \cup K') \cap L$.

Oct/Nov 2022

$\mathcal{E} = \{\text{students in a class}\}$ $C = \{\text{students who play cricket}\}$ $F = \{\text{students who play football}\}$ Altogether, the class has 36 students. 15 students play cricket. 20 students play football. The Venn diagram has 11 in $C$ only and 5 outside both sets.

Oct/Nov 2023

The universal set $\mathcal{E} = \{2, 4, 8, 9, 10, 12\}$ is given, with $Q = \{\text{square numbers}\}$ and $R = \{\text{multiples of }4\}$. The diagram of the Venn diagram for sets $Q$ and $R$ within $\mathcal{E}$ is shown.

Oct/Nov 2023

Determine $n(A \cap B)$.

Oct/Nov 2023

The figure shows two Venn diagrams, each one placed inside a rectangle that represents the universal set.

Oct/Nov 2023

Jo asks a group of people whether they own a car ($C$) and whether they own a motorbike ($M$). There are 86 people who own a car. There are 39 people who own a motorbike. 7 people own neither a car nor a motorbike. A Venn diagram is displayed with $C$ and $M$, and 74 appears in region $C$ only.

Oct/Nov 2024

$\mathcal{E} = \{\text{office workers}\}$, $C = \{\text{workers who drink coffee}\}$, $T = \{\text{workers who drink tea}\}$. 47 people work in the office. 32 people drink tea. A Venn diagram is displayed. The region outside both circles contains 8. The T-only region contains 6.

Oct/Nov 2024

$\mathcal{E}=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ $P=\{\text{odd numbers}\}$ $Q=\{\text{multiples of } 3\}$ $R=\{\text{square numbers}\}$

Oct/Nov 2024

$\mathcal{E} = \{8 \times 10^{-1},\; 0.8,\; 8\%,\; \sqrt{0.08}\}$ $A = \{a : 0.08 < a \le 0.8\}$ $B = \{b : b \ge 0.8\}$ The Venn diagram for sets $A$ and $B$ within the universal set $\mathcal{E}$ is displayed.

Oct/Nov 2024

The Venn diagram indicates how many elements lie in each region. The region counts are as follows: in A only: 1; in A and B only: 3; in B only: 5; in A and C only: 1; in A, B and C: 5; in B and C only: 2; in C only: 6; outside all sets: 2. The shaded part includes B only, B and C only, and C only.

Oct/Nov 2024

Complete the Venn diagram shown.

Oct/Nov 2024

E = {students in a year group}, H = {students who study History}, G = {students who study Geography}. There are 80 students in the year group altogether. 40 students study History. The Venn diagram shows 10 in the History-only region and 15 outside both sets.

Oct/Nov 2025

The universal set is $\xi = \{1,2,3,4,5,6,7,8,9,10\}$, with $A=\{1,3,5,7,9\}$ and $B=\{1,3,6,10\}$.

Oct/Nov 2025

In part (a), a Venn diagram is shown for sets A and B, and the overlapping section is shaded. Part (b): E = {people in a club}, T = {people who play tennis}, S = {people who go swimming}. The club has 60 people in total. 36 people go swimming. The Venn diagram shows 10 outside both sets and 30 in the S-only region.

Oct/Nov 2025

$\mathcal{E}=\{n:n\text{ is an integer and }1\le n\le 8\}$, $A=\{\text{values that are factors of }12\}$, $B=\{\text{odd numbers}\}$.

Oct/Nov 2025

Indicate the shaded area on each Venn diagram.

Oct/Nov 2025

The pupils are asked whether they prefer football (F) or rugby (R). The Venn diagram displays the outcomes.

Oct/Nov 2025