Sets & Venn Diagrams Difficulty: Hard

(1)

Head to savemyexams.co.uk for more awesome resources

Model Answers 2

Sets & Venn Diagrams

Difficulty: Hard

Time allowed:

45 minutes

Score:

/35

Percentage:

/100

Grade Boundaries:

CIE IGCSE Maths (0580)

CIE IGCSE Maths (0980)

A*

A

B

C

D

E

>88%

76%

63%

51%

40%

30%

9

8

7

6

5

4

3 >94%

85%

77%

67%

57%

47%

35%

Level

IGCSE

Subject

Maths (0580/0980)

Exam Board

CIE

Topic

Number

Sub-Topic

Sets & Venn Diagrams

Paper

Paper 2

Difficulty

Hard

(2)

A = {square numbers} B = {1, 2, 3, 4, 5, 6}

(a) Write all the elements of in their correct place in the Venndiagram. [2]

(b) List the elements of (A ∪B)'.

(c) Find n(A ∩ B'). [1]

[1]

𝟕𝟕, 𝟖𝟖, 𝟏𝟏𝟏𝟏

1

(3)

Head to savemyexams.co.uk for more awesome resources A B y x 12 – x 15 – x 20 – x 14 13 8 C

The Venn diagram shows the number of elements in sets A, B and C. (a) n(A ∪ B ∪ C ) = 74 Find x. (b) n( ) = 100 Find y. [2] [1]

𝑛𝑛(𝐴𝐴 ⋃ 𝐵𝐵 ⋃ 𝐶𝐶) is the sum of all elements in the sets A, B and C.

13 + 14 + 8 + 12 − 𝑥𝑥 + 20 − 𝑥𝑥 + 15 − 𝑥𝑥 + 𝑥𝑥 = 74

82 − 2𝑥𝑥 = 74

2𝑥𝑥 = 8

𝒙𝒙 = 𝟒𝟒

𝑛𝑛 𝜉𝜉 is the sum of all elements in and outside of the sets.

𝑦𝑦 + 74 = 100

(4)

(c) Find the value of n((A ∪ B)' ∩ C). [1]

𝑛𝑛((𝐴𝐴 ∪ 𝐵𝐵)′ ∩ 𝐶𝐶)

describes the elements outside the union of A and B which

intersect with C

(5)

3 (a)

A B

Shade the region A ∩ B'. [1]

𝐴𝐴 ∩ 𝐵𝐵′is a region which belongs both A and does not belong to B. (It belongs strictly only to A)

(b)

A B

7 4 5

3

This Venn diagram shows the number of elements in each region.

Write down the value of n (A∪ B'). [1]

𝐴𝐴 ∪ 𝐵𝐵′is a region which belongs to either A or does not belong to B.

By summing the elements in the shaded regions:

𝑛𝑛 𝐴𝐴 ∪ 𝐵𝐵′ _{= 3 + 7 + 4}

(6)

In a survey of 60 cars, 25 use diesel, 20 use liquid hydrogen and 22 use electricity. No cars use all three fuels and 14 cars use both diesel and electricity.

There are 8 cars which use diesel only, 15 cars which use liquid hydrogen only and 6 cars which use electricity only.

In the Venn diagram below = {cars in the survey}, D = {cars which use diesel},

L = {cars which use liquid hydrogen}, E = {cars which use electricity}.

8 6 15 ... ... ... ... ... D E L

(a) Use the information above to fill in the five missing numbers in the Venn diagram. [4]

60 elements in total.

25 in D total, 20 in L total, 22 in E total.

14 in D

∩ E and 0 in D ∩ E ∩ L.

(7)

14

0

3

2

12

(b) Find the number of cars which use diesel but not electricity. _[1]

8 + 3

= 𝟏𝟏𝟏𝟏

(c) Find n(D'∩ (E ∪ L)). [1]

The number of elements in the intersection of the union of E and L with the

elements not in D.

(8)

In a group of 30 students, 18 have visited Australia, 15 have visited Botswana and 5 have not visited either country.

Work out the number of students who have visited Australia but not Botswana. [2]

We have an overlap of

18 + 15 − (30 − 5)

= 8

Hence, 8 students have visited both countries.

So, we have

18 − 8

= 𝟏𝟏𝟏𝟏

(9)

In a group of 24 students, 21 like football and 15 like swimming. One student does not like football and does not like swimming.

Find the number of students who like both football and swimming. [2]

We have

24 − 1 = 23

students that like either one or the other or both.

We have an overlap of

(10)

Shade the region required in each Venn Diagram. A

C

B A B

A∩ B ∩ C A ∪ B′ [2]

We shade the intersection of all 3 Sets A, B and C, therefore, the area which is common

to all 3 sets.

B’ – the complement of B, represents all the elements which are not in Set B.

For the reunion of B’ with Set A, we shade therefore every region except Set B, including the

intersection between Set A and B since that is considered Set A.

(11)

12 A and B are sets.

Write the following sets in their simplest form.

(a) A

∩

A'. _[1]

(b) A

∪

A'. [1]

(c) (A

_∩

B)

∪

(A

∩

B' ). [1]

∅ - the empty set

𝜉𝜉 – all the elements

(12)

Head to savemyexams.co.uk for more awesome resources n(A) = 18, n(B) = 11 and n(A ∪ B)′ = 0.

(a) Label the Venn diagram to show the sets A and B where n(A ∪ B) =18.

Write down the number of elements in each region. [2]

(b) Draw another Venn diagram to show the sets A and B where n(A ∪ B) = 29.

Write down the number of elements in each region. [2]

The union of A and B is 18.

This represents all the elements which are either in Set A, B and both.

The complement of their union is 0, meaning that there are no

elements which are not in either Set A, B and both.

The union of the 2 sets needs to be 29.

(13)

11 Write each of these four numbers in the correct place in the Venn Diagram below.

4 ,

_√

12, 112 2.6, 17

√

7 Rational numbers Integers [4]

12 – irrational number (non-ending, non-recurring decimal)

112

7

= 4 – integer

4

17

= 0.23 and 2.6 – rational numbers

(14)

Three sets A, B and K are such that A ⊂ K, B ⊂ K and A ∩ B = Ø.

Draw a Venn diagram to show this information. [2]