Sets (Cambridge International IGCSE Maths): Exam Questions

$E$ = {21, 22, 23, 24, 25, 26, 27, 28, 29, 30}

$A$ = {22, 24, 26, 28, 30}

$B$ = {21, 24, 27, 30}

List the members of the set

i) $A\cap B$

ii) $A'$

Extract:

![A] = {22, 24, 26, 28, 30}

![B] = {21, 24, 27, 30}

$C$ = {23, 25, 29}

Using set notation, find an expression for $C$ in terms of $A$ and $B$.

$E$= {letters of the alphabet} $B$ = {b, r, a, z, i, l} $I$ = {i, r, e, l, a, n, d}

List the members of the set

i) $B\cup I$

ii) $B\cap Iʹ$

Extract:

![B] = {b, r, a, z, i, l}

$K$ = {k, e, n, y, a}

Cody writes down the statement $B\cap K=\emptyset$

Cody’s statement is wrong. Explain why.

$\begin{array}{rcl} B = \{ \mathrm{b} , \mathrm{l} , \mathrm{u} , \mathrm{e} \} \\ G = \{ \mathrm{g} , \mathrm{r} , \mathrm{e} , \mathrm{y} \} \\ W = \{ \mathrm{w} , \mathrm{h} , \mathrm{i} , \mathrm{t} , \mathrm{e} \} \end{array}$

List all the members of the set

i) $B\cup G$

ii) $W\cap G'$

Extract:

B = {b, l, u, e} G = {g, r, e, y} W = {w, h, i, t, e}

Serena writes down the statement $B\cap G\cap W=\emptyset$

Is Serena’s statement correct? You must give a reason for your answer.

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 = { x colon 41 <= x <= 50 }

A = left curly bracket x colon xis an odd numberright curly bracket

B = left curly bracket x colon xis a multiple of 3right curly bracket

C = left curly bracket x colon xis a prime numberright curly bracket

Complete the Venn diagram to show this information.

Extract:

ℰ equals {x : 41 ≤ x ≤ 50} A = {x : x is an odd number} B = {x : x is a multiple of 3} C = {x : x is a prime number}

List the elements of

i) $A\cap C,$

ii) $( B \cup C ) '.$

Extract:

![x] is an integer.

![ℰ equals open curly brackets x space colon space 41 less or equal than x less or equal than 50 close curly brackets]

![A equals left curly bracket x space colon space x space]is an odd number![right curly bracket]

![B equals left curly bracket x space colon space x space]is a multiple of 3![right curly bracket]

![C equals left curly bracket x space colon space x space]is a prime number![right curly bracket]

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

$E=${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.

Extract:

There are 240 students in the school.
• 120 students play football • 40 students play baseball
• 90 students play football but not baseball.

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

Use set notation to complete the statements.

i) $d$...................$A$

ii) left curly bracket f , g right curly bracket =...............................

iii) Complete the statement.

$n(.........................)=6$ [1]

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

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.

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

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

Find

n(A ∩ B)

Extract:

n(E) = 20, n((A ∪ B)′) = 3, n(A) = 10 and n(B) = 13.

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?

ii) How many students study exactly two subjects?

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

calligraphic E equals left curly bracket x space colon space x spaceis an integer and 1 <= x <= 10 right curly bracket A equals left curly bracket x space colon space x spaceis evenright curly bracket $4 \text{element} of A \cap B$ $\mathrm{n} ( A \cap B ) = 1$ left parenthesis A union B right parenthesis apostrophe equals left curly bracket 1 comma 7 comma 9 right curly bracket

Complete the Venn diagram using this information.

Extract:

![calligraphic E equals left curly bracket x space colon space x space]is an integer and ![1 less or equal than x less or equal than 10 right curly bracket]

![A equals left curly bracket x space colon space x space]is even![right curly bracket]

![4 element of A intersection B straight n left parenthesis A intersection B right parenthesis equals 1 left parenthesis A union B right parenthesis apostrophe equals left curly bracket 1 comma 7 comma 9 right curly bracket]

Use your Venn diagram to complete the statement.

B equals left curly bracket....................................................right curly bracket

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 $n(T\cap(E\cup S))$.

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

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

Find

$n(A\cup B)$

$n(A\cap B)$

Extract:

The Venn diagram shows a universal set and three sets A, B and C. 6, 3, 8, 2, 5 and 4 represent the numbers of elements.

$n(B\cap C')$

Extract:

The Venn diagram shows a universal set ![calligraphic E] and three sets ![A comma space B] and ![C].

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

$n(A'\cup B'\cup C')$

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

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

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

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

Complete the Venn diagram for this information.

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.

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

Complete the Venn diagram.

Extract:

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.

n(P ∩ T) = 8

n(B ∩ T) = 3

n(B ∩ P) = 6

Find $n((P\cup B)')$.

Extract:

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

Use set notation to complete the statement.

$(P\cup T\cup B)'=$...........................

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 $x=4$.

ii) Complete the Venn diagram.

Extract:

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.

i) Use set notation to complete this statement.

$( G \cup M \cup H ) ' =$....................... [1]

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

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.

Extract:

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

Given that a total of 50 students answered the question,

work out the value of $x$.

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

Extract:

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

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.

Extract:

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.

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