Set Notation & Venn Diagrams (Cambridge (CIE) O Level Maths): Exam Questions

There are 32 students in a class.

5 do not study any languages.
15 study German (G).
18 study Spanish (S).

Complete the Venn diagram to show this information.

$x$ is an integer.

$ℰ=\left\{x:41\le x\le 50\right\}$

$A=\{x:x$is an odd number$\}$

$B=\{x:x$is a multiple of 3$\}$

$C=\{x:x$is a prime number$\}$

Complete the Venn diagram to show this information.

List the elements of

i) $A\cap C,$  

[1]

ii) $\left(B\cup C\right)\text{'}.$

[1]

Find $\mathrm{n}\left(A\cap B\cap C\right).$

$\mathcal{E}=${students in a school}
$F=${students who play football}
$B=${students who play baseball}
 
There are 240 students in the school.
 
• 120 students play football
• 40 students play baseball 
• 90 students play football but not baseball.

Complete the Venn diagram to show this information.

Find $\text{n }(F\text{'}\cap B\text{'}).$

Use set notation to complete the statements.

i) $d$...................$A$  

[1]

ii) $\{f,g\}=$...............................  

[1]

iii) Complete the statement.  

$\mathrm{n}(.........................)=6$ [1]

In the Venn diagram below, shade $C\cap D\text{'}.$

The number of members of a leisure centre using the exercise machines (E), the swimming pool (S) and the tennis courts (T) is shown on the Venn diagram.

i) Find the number of members using only the tennis courts.  

[1]

ii) Find the number of members using the swimming pool.

[1]

$\mathrm{n}\left(\mathcal{E}\right)=20,\mathrm{n}(A\cup B)\text{'}=3,\mathrm{n}\left(A\right)=10\mathrm{and}\mathrm{n}\left(B\right)=13$.
The Venn diagram shows some of this information.

Find

n(A ∩ B)

n(A′ ∩ B) .

  The Venn diagram above shows information about the number of students who study Music (M), Drama (D) and Geography (G).

i) How many students study Music?  

[1]

ii) How many students study exactly two subjects?

[1]

In the Venn diagram above, shade  $M\cap D\text{'}$.

$\mathcal{E}=\{x:x$is an integer and $1\le x\le 10\}$ $A=\{x:x$is even$\}$ $$\begin{gathered} 4\in A\cap B \\ \mathrm{n}(A\cap B)=1 \\ (A\cup B)\text{'}=\{1,7,9\} \end{gathered}$$  

Complete the Venn diagram using this information.

Use your Venn diagram to complete the statement.

$B=\{$....................................................$\}$

The number of members of a leisure centre using the exercise machines (E), the swimming pool (S) and the tennis courts (T) is shown on the Venn diagram.

Find $\mathrm{n}(T\cap (E\cup S\left)\right)$.

The Venn diagram shows a universal set $\mathcal{E}$ and three sets $A,B$ and $C$.

6, 3, 8, 2, 5 and 4 represent the numbers of elements.

Find

$\mathrm{n}(A\cup B)$

$\mathrm{n}(A\cap B)$

$\mathrm{n}(B\cap C\text{'})$

$\mathrm{n}(A\text{'}\cup B\text{'}\cup C\text{'})$

$\mathcal{E}$ = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

$A\cap B$ = {5, 10, 15}

$B\text{'}$= {7, 8, 9, 11, 12, 13, 14}

$A\text{'}$= {4, 6, 7, 8, 14}

Complete the Venn diagram for this information.

The sets $A$, $B$ and $C$ are shown in the Venn diagram.

$$\begin{gathered} \mathcal{E}=\left\{x:x\mathrm{is}\mathrm{an}\mathrm{integer},1\le x\le 18\right\} \\ A=\left\{x:xisanevennumber\right\} \\ B=\left\{x:xisamultipleof5\right\} \end{gathered}$$

i) Find $\mathrm{n}(A\cup B)$.

[1]

ii) (a) Given that $A\cap B\text{'}\cap C\text{'}=\left\{2,6,14,18\right\},$list the members of $C$.

[1]

(b) Describe the set $C$ in words.

$C$ = {$x:x$ is ........................... } [1]

This year, 40 students have each travelled by one or more of plane ($P$), train ($T$) or boat ($B$).

7 have travelled only by plane.
11 have travelled only by train.
9 have travelled only by boat.

$\mathrm{n}(P\cap T)=8$
$\mathrm{n}(B\cap T)=3$
$\mathrm{n}(B\cap P)=6$

Complete the Venn diagram.

Find $\mathrm{n}\left(\right(P\cup B)\text{'})$.

Use set notation to complete the statement.

$(P\cup T\cup B)\text{'}=$........................... 

i) Use set notation to complete this statement.  

$\left(G\cup M\cup H\right)\text{'}=$....................... [1]

ii) Find $\mathrm{n}\left(G\cap \left(M\cup H\right)\right)$.  

[1]

50 students study at least one of the subjects geography ($G$), mathematics ($M$) and history ($H$).
18 study only mathematics.
19 study two or three of these subjects.
23 study geography.
 
The Venn diagram below is to be used to show this information.

i) Show that $x=4$.  

[2]

ii) Complete the Venn diagram.  

[2]

Some students in a school were asked the following question.

“Do you have a dog ($D$), a cat ($C$) or a rabbit ($R$)?”

Of these students

   28 have a dog
   18 have a cat
   20 have a rabbit
    8 have both a cat and a rabbit
    9 have both a dog and a rabbit
    $x$ have both a dog and a cat
    6 have a dog, a cat and a rabbit
    5 have not got a dog or a cat or a rabbit

Using this information, complete the Venn diagram to show the number of students in each appropriate subset.
Give the numbers in terms of $x$ where necessary.

Given that a total of 50 students answered the question,

work out the value of $x$.

$x$ = ........................

Find  $n(Cʹ\cap Dʹ)$

Some students were asked the following question.

“Which of the subjects Russian (R), French (F) and German (G) do you study?”

Of these students

4 study all three of Russian, French and German
10 study Russian and French
13 study French and German
6 study Russian and German
24 study German
11 study none of the three subjects
the number who study Russian only is twice the number who study French only.

Let $x$ be the number of students who study French only.

Show all this information on the Venn diagram, giving the number of students in each appropriate subset, in terms of $x$ where necessary.

Given that the number of students who were asked the question was 80, work out the number of these students that study Russian.

A and B are two sets.

n(ξ) = 37

n(A) = 22

n(A ∩ B) = 12

n(A ∪ B) = 30

Complete the Venn diagram to show the number of elements in each region.

Find

(i)  n(A ∩ B′)

[1]

(ii) n(A′ ∪ B′)

[1]

$\mathcal{E}$ = { A, C, E, G, H, J, N, R, T, Z }

$W$ = { $x:x$ has rotational symmetry of order 2 }
$X$ = {$x:x$ has line symmetry }

$Y$ = { R, A, N, G, E }

i) Complete the Venn diagram.

[3]

ii) List the elements of $X\cap \left(W\cup Y\right)\text{'}$.

iii) Find  $\mathrm{n}\left(W\cup X\cup Y\right)\text{'}$.

[1]

iv) Using set notation, complete this statement. 

................................... = $\varnothing$[1]