1 (CIE 2012, s, paper 21, question 6)

By shading the Venn diagrams below, investigate whether each of the following statements is true or false. State your conclusions clearly.

(i) $\D A\cap B = (A'\cap B)'$ [2]

(ii) $\D X\cap Y = X'\cup Y'$ [2]

(iii) $\D (P\cap Q)\cup (Q\cap R) = Q\cap (P\cup R)$ [2]

2 (CIE 2012, s, paper 22 , question 1 )

It is given that $P$ is the set of prime numbers, $S$ is the set of square numbers and $N$ is the set of numbers between 10 and 90 . Write each of the following statements using set notation.

(i) 7 is a prime number.

(ii) 8 is not a square number.

(iii) There are 6 square numbers between 10 and 90.

(a) On the Venn diagrams below, shade the region corresponding to the set given below each Venn diagram.

(b) It is given that sets $\D E, B, S$ and $\D F$ are such that

$\D E$ = {students in a school},

$\D B =$ {students who are boys},

$\D S =$ {students in the swimming team},

$\D F =$ {students in the football team}.

Express each of the following statements in set notation.

(i) All students in the football team are boys. [1]

(ii) There are no students who are in both the swimming team and the football team. [1]

4 (CIE 2012, w, paper 21, question 2)

(a) It is given that $\D E$ is the set of integers, $\D P$ is the set of prime numbers between 10 and 50, $\D F$ is the set of multiples of 5, and $\D T$ is the set of multiples of 10. Write the following statements using set notation.

(i) There are 11 prime numbers between 10 and 50. [1]

(ii) 18 is not a multiple of 5. [1]

(iii) All multiples of 10 are multiples of 5. [1]

(b) (i) In the Venn diagram below shade the region that represents $\D (A'\cap B) \cup (A\cap B').$ [1]

(ii) In the Venn diagram below shade the region that represents $\D Q\cap (R\cup S' ).$ [1]

5 (CIE 2013, s, paper 12, question 1)

The Venn diagram shows the universal set $\D E$, the set $\D A$ and the set $\D B.$ Given that $\D n(B ) = 5, n(A') = 10$

and $\D n(E) = 26,$ find

(i) $\D n(A \cap B),$ [1]

(ii) $\D n(A),$ [1]

(iii) $\D n(B' \cap A).$ [1]

6 (CIE 2013, s, paper 21, question 9)

It is given that $\D x \in R$ and that

$\D E = {x : − 5 < x < 12},$

$\D S = {x : 5x + 24 > x^2},$

$\D T = {x : 2x + 7 > 15}.$

Find the values of $\D x$ such that

(i) $\D x \in S ,$ [3]

(ii) $\D x \in S\cup T ,$ [2]

(iii) $\D x \in (S\cap T )'.$ [3]

7 (CIE 2013, w, paper 11, question 4)

The sets $\D A$ and $\D B$ are such that

$\D A=\{ x:\cos x=\frac{1}{2} , 0^{\circ}\le x\le 620^{\circ},\}$

$\D B=\{ x:\tan x=\sqrt{3} , 0^{\circ}\le x\le 620^{\circ},\}$

(i) Find $\D n(A).$ [1]

(ii) Find $\D n(B).$ [1]

(iii) Find the elements of $\D A \cup B.$ [1]

(iv) Find the elements of $\D A \cap B.$ [1]

8 (CIE 2013, w, paper 23, question 5)

(a) (i) In the Venn diagram below shade the region that represents $\D (A\cup B)' $ [1]

(ii) In the Venn diagram below shade the region that represents $\D P\cap Q\cap R'.$ [1]

(b) Express, in set notation, the set represented by the shaded region. [1]

(c) The universal set $\D E$ and the sets $\D V$ and $\D W$ are such that $\D n(E) = 40, n(V ) = 18$ and $\D n(W) = 14.$ Given that $\D n(V\cap W) = x$ and $\D n((V\cup W)') = 3x$ find the value of $\D x.$

You may use the Venn diagram below to help you. [3]

9 (CIE 2014, s, paper 11, question 3)

(a) On the Venn diagrams below, shade the regions indicated. [3]

(b) Sets $\D P$ and $\D Q$ are such that $\D P=\{x:x^2+2x=0\},$ and $\D Q=\{x:x^2+2x+7=0\},$ where $\D x\in R.$

(i) Find $\D n(P).$ [1]

(ii) Find $\D n(Q).$ [1]

10 (CIE 2014, s, paper 12, question 2)

(a) On the Venn diagrams below, draw sets $\D A$ and $\D B$ as indicated.

(i)

[2]

(b) The universal set $\D E$ and sets $\D P$ and $\D Q$ are such that $\D n(E) = 20, n(P \cup Q) = 15, n(P) = 13$ and $\D n(P \cap Q) = 4.$ Find

(i) $\D n(Q),$ [1]

(ii) $\D n(P \cup Q)',$ [1]

(iii) $\D n(P \cap Q').$ [1]

11 (CIE 2014, s, paper 23, question 4)

(a) Illustrate the following statements using the Venn diagrams below.

(i) $\D A \cup B = A $ (ii) $\D A \cap B \cap C = \emptyset.$ [2]

(i) 50 is not a cube number. [1]

(ii) 64 is both a square number and a cube number. [1]

(iii) There are 90 integers between 1 and 100 inclusive which are not square numbers. [1]

12 (CIE 2014, w, paper 13, question 3)

The universal set $\D E$ is the set of real numbers. Sets $\D A, B$ and $\D C$ are such that

$\D A = \{x:x^2+5x+6=0\}, $

$\D B = \{x:(x-3)(x+2)(x+1)=0\}, $

$\D C = \{x:x^2+x+3=0\}.$

(i) State the value of each of $\D n(A), n(B)$ and $\D n(C).$ [3]

(ii) List the elements in the set $\D A \cup B.$ [1]

(iii) List the elements in the set $\D A \cap B.$ [1]

(iv) Describe the set $\D C'.$ [1]

13 (CIE 2014, w, paper 21, question 1)

(a) On each of the Venn diagrams below shade the region which represents the given set.

(b) In a year group of 98 pupils, $\D F$ is the set of pupils who play football and $\D H$ is the set of pupils who play hockey. There are 60 pupils who play football and 50 pupils who play hockey. The number that play both sports is $\D x$ and the number that play neither is $\D 30 - 2x.$ Find the value of $\D x.$ [3]

The Venn diagram above shows the sets $A, B$ and $C$. It is given that $\mathrm{n}(A \cup B \cup C)=48$, $\mathrm{n}(A)=30, \quad \mathrm{n}(B)=25, \quad \mathrm{n}(C)=15$, $\mathrm{n}(A \cap B)=7, \quad \mathrm{n}(B \cap C)=6, \quad \mathrm{n}\left(A^{\prime} \cap B \cap C^{\prime}\right)=16 .$

(i) Find the value of $x$, where $x=\mathrm{n}(A \cap B \cap C)$.

(ii) Find the value of $y$, where $y=\mathrm{n}\left(A \cap B^{\prime} \cap C\right)$.

(iii) Hence show that $A^{\prime} \cap B^{\prime} \cap C=\varnothing$.

15 (CIE $2015, \mathrm{~s}$, paper 22, question 1$)$

The universal set contains all the integers from 0 to 12 inclusive. Given that $A=\{1,2,3,8,12\}, \quad B=\{0,2,3,4,6\} \quad$ and $C=\{1,2,4,6,7,9,10\}$

(ii) state the value of $\mathrm{n}\left(A^{\prime} \cap B^{\prime} \cap C\right)$, $[1]$

(iii) write down the elements of the set $A^{\prime} \cap B \cap C$. $[1]$

16 (CIE $2015, \mathrm{w}$, paper 11, question 6 )

It is given that $\mathscr{E}=\{x: 1 \leqslant x \leqslant 12$, where $x$ is an integer $\}$ and that sets $A, B, C$ and $D$ are such that $A=\{$ multiples of 3$\}$, $B=\{$ prime numbers $\}$, $C=\{$ odd integers $\}$ $D=\{$ even integers $\}$. Write down the following sets in terms of their elements.

(i) $A \cap B$

(ii) $A \cup C$ $[1]$

(iii) $A^{\prime} \cap C$ $[1]$

(iv) $(D \cup B)^{\prime}$ $[1]$

(v) Write down a set $E$ such that $E \subset D$. [1]

18 (CIE 2016, march, paper 22, question 2)

The sets $A, B$ and $C$ are such that $B \cap C=\varnothing$ $\mathrm{n}(A \cap B)=2$ $\mathrm{n}(B)=12$ $\mathrm{n}(B \cup C)=14$$\mathrm{n}(A \cup B)=19$ Complete the Venn diagram to show the sets $A, B$ and $C$ and hence state $\mathrm{n}\left(A \cap B^{\prime} \cap C^{\prime}\right)$.

19 (CIE 2016, s, paper 12, question 1 )

(a) The universal set $\mathscr{\text { is the set of real numbers and sets }} X, Y$ and $Z$ are such that $X=\{$ integer multiples of 5$\}$ $Y=$ \{integer multiples of 10$\},$ $Z=\{\pi, \sqrt{2}, \mathrm{e}\}$

Use set notation to complete the two statements below.

$\begin{array}{ll}x & Y \cap Z=\end{array}$

20 (CIE 2016, s, paper 21, question 2)

(a) Illustrate the statements $A \subset B$ and $B \subset C$ using the Venn diagram below.

(b) It is given that the elements of set $\mathscr{E}$ are the letters of the alphabet, the elements of set $P$ are the letters in the word maths, the elements of set $Q$ are the letters in the word exam.

(i) Write the following using set notation.

The letter $h$ is in the word maths.

(ii) Write the following using set notation.

The number of letters occurring in both of the words maths and exam is two. $[1]$

(iii) List the elements of the set $P \cap Q^{\prime} .$

$[1]$

21 (CIE 2016, w, paper 11, question 1)

(a) Sets $\mathscr{E}, A$ and $B$ are such that $$ \mathrm{n}(\mathscr{E})=26, \mathrm{n}\left(A \cap B^{\prime}\right)=7, \mathrm{n}(A \cap B)=3 \text { and } \mathrm{n}(B)=15 \text { . } $$ Using a Venn diagram, or otherwise, find

(i) $\mathrm{n}(A)$

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

(iii) $\mathrm{n}(A \cup B)^{\prime}$.

(b) It is given that $8=\{x: 0<x<30\}, P=\{$ multiples of 5$\}, Q=\{$ multiples of 6$\}$ and $R=\{$ multiples of 2$\}$. Use set notation to complete the following statements.

(i) $Q \ldots \ldots \ldots . R$,

(ii) $P \cap Q=\ldots \ldots \ldots$..

22 (CIE 2017, march, paper 12, question 1)

(a) It is given that $\mathscr{=}\{x: 0<x<35, x \in \mathbb{R}\}$ and sets $A$ and $B$ are such that $A=\{$ multiples of 5 ; and $B=\{$ multiples of 7$\}$

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

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

(b) It is given that sets $X, Y$ and $Z$ are such that

$$X \cap Y=Y, \quad X \cap Z=Z \text { and } Y \cap Z=\emptyset \text { . } $$ On the Venn diagram below, illustrate sets $X, Y$ and $Z$.

23 (CIE 2017, s, paper 12 , question 1)

On each of the Venn diagrams below, shade the region which represents the given set. $(A \cup B) \cap C$ [3]

24 (CIE 2017, s, paper 13, question 1)

(a) On the Venn diagram below, shade the region which represents $\left(A \cap B^{\prime}\right) \cup\left(C \cap B^{\prime}\right)$. $[1]$

(b) Complete the Venn diagram below to show the sets $Y$ and $Z$ such that $Z \subset X \subset Y$. [1]

25 (CIE 2017, s, paper 21, question 7)

(b) In a group of students, each student studies at most two of art, music and design. No student studies both music and design.

$A$ denotes the set of students who study art, $M$ denotes the set of students who study music, $D$ denotes the set of students who study design.

(i) Write the following using set notation.

No student studies both music and design. There are 100 students in the group. 39 students study art, 45 study music and 36 study design. 12 students study both art and music. 25 students study both art and design.

(ii) Complete the Venn diagram below to represent this information and hence find the number of students in the group who do not study any of these subjects.

27 (CIE 2017, w, paper 12 , question 1)

28 (CIE 2017, w, paper 23 , question 1)

(a) On each of the diagrams below, shade the region which represents the given set. $(A \cup B) \cap C^{\prime} \quad\left(A \cap B^{\prime}\right) \cup C \quad[2]$

(b) The Venn diagram shows the number of elements in each of its subsets. Complete the following.

$\mathrm{n}\left(P^{\prime}\right)=\ldots \ldots \ldots \ldots$ $\mathrm{n}((Q \cup R) \cap P)=\ldots \ldots \ldots \ldots$ $\mathrm{n}\left(Q^{\prime} \cup P\right)=\ldots \ldots \ldots \ldots$ $[3]$

$\mathrm{n}\left(P^{\prime}\right)=\ldots \ldots \ldots \ldots$ $\mathrm{n}((Q \cup R) \cap P)=\ldots \ldots \ldots \ldots$ $\mathrm{n}\left(Q^{\prime} \cup P\right)=\ldots \ldots \ldots \ldots$ $[3]$

(a) Using set notation, write down the set represented by the shaded region.

(b) $\begin{aligned} 8 &=\{1,2,3,4,5,6,7,8,9,10\} \\ A &=\{x: x \text { is a prime number }\} \\ B &=\{x: x \text { is an cven number }\} \\ C &=\{1,2,3,4,8\} \end{aligned}$

(ii) Write down the value of $\mathrm{n}(A \cup B \cup C)^{\prime}$ ). $[1]$

30 (CIE 2018, s, paper 21, question 1) $A, B$ and $C$ are subsets of the same universal set.

(i) Write each of the following statements in words.

(a) $A \not \subset B$ $[1]$

(b) $A \cap C=\varnothing$ $[1]$

(ii) Write each of the following statements in set notation.

(a) There are 3 elements in set $A$ or $B$ or both.

(b) $x$ is an element of $A$ but it is not an element of $C$. $[\mathrm{l}]$

31 (CIE 2018, s, paper 22 , question 2)

(b) The universal set $\varepsilon$ and sets $P, Q$ and $R$ are such that $(P \cup Q \cup R)^{\prime}=\varnothing$, $P^{\prime} \cap(Q \cap R)=\varnothing$,$\mathrm{n}(Q \cap R)=8$ $\mathrm{n}(P \cap R)=8$, $\mathrm{n}(P \cap Q)=10$ $\mathrm{n}(P)=21, \quad \mathrm{n}(Q)=15, \quad \mathrm{n}(\mathscr{E})=30 .$ Complete the Venn diagram to show this information and state the value of $n(R)$. $[4]$

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