Sets & Number Theory
Sets
IGCSE MATHEMATICS
Sets :
A set is a well defined group of object or symbol. The object or symbol are called the element of the
set .
Write the another two element of the set in the following :1) { Asia , Africa , Europe , ………….. }
2) { 2 , 4 , 6 , …………………. }
3) { Sunday , Monday , Tuesday ………….. }
4) { 1, 3, 6, 10 ………………….}
5) { a, e, i …………….. }
6) { Earth, Mars, venus, …………………}
Subsets: If all the element of one set X are also the element of another set Y, then X is said to be a subset of Y
This is written as X .
1) A = 1 2 3 4 5 6 7 8 9 10
(i)
List subset B ( even numbers )
(ii)
List subset C ( prime numbers )
2) P = whole number less than 30
a) List the subset Q ( even numbers )
b) List the subset R ( odd numbers )
c) List the subset S ( prime numbers )
d) List the subset T( square numbers )
3) P = whole number between 50 and 70
a) List the subset B ( multiples of 5 )
b) List the subset C ( multiples of 3 )
c) List the subset D( Square numbers )
Page 1 of 9
IGCSE MENTORS
Sets & Number Theory
Venn diagrame
1) If A = a b c d e f g h i j k l m
B = cdelmnopqrs
Represent onto a venn diagrame
A
B
n
a b f cd
g h
e l
i j k m
o
p
q
r
s
2) If  = 1 2 3 4 5 6 7 8 9 10 11 12
A = 1 2 5 6 11 12
B = 7 8 9 10 11 12
Represent onto a venn diagrame

A
1
B
2
11
7
5
8
9
6
12
10
3
4
3. Answer the following with the help of venn diagrame
(i) A U B =
(ii) A ∩ B =
(iii) A' U B =
(iv) A' ∩ B =
(v) A' U B' =
(vi) A' ∩ B' =
(vii) ( AUB )' =
(viii) ( A∩B )' =
A

1
B
2
7
8
3
9
14
10
4
11
5
12
6
13
15
16
Page 2 of 9
IGCSE MENTORS
Sets & Number Theory
4. In a class of 31 students some study physics some study chemistry. If 22 study physics,
20 study chemistry and 5 study neither , calculate the number of student who take both
the subjects.
5. In a class of 35 students, 19take Spanish, 18 take French and 3 take neither . calculate how
many take :a) both French and Spanish
b) just Spanish
c) just French
6. In a group of 108 students 60 like football 53 like tennis and 10 like neither , Calculate
the number of student who like football but not tennis.
7. 46 tourists took a holiday together in Switzerland and Denmark. 30 had already visited
Switzerland and 28 had already visited Denmark but 4 visited neither of the place.
Calculate:i) The number of tourists who visited both Switzerland and Denmark
ii) the number of tourists who visited only Switzerland
iii) the number of tourists who visited Denmark but not Switzerland
8. In a class of 30 students, 14 take physics, 16 take chemistry and 4 take neither.
a) Enter the above information in a Venn diagram
b) Form an equation in x
c) Find the value of x
d) Hence find how many takes only physics
9. In a class of 33 students, 19 take Dhivehi, 16 take Hindi and 4 take neither.
a. Represent the above information on a Venn-diagram.
b. Calculate how many take both Dhivehi and Hindi.
c. Calculate how many take only Dhivehi.
Shading
Shade the region as mentioned below and copy the figure for each of the question.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
AUB
A∩B
A' U B
A ∩B'
(AUB)'
(A∩B)'
A' U B'
A' ∩ B'
A
B
Page 3 of 9
IGCSE MENTORS
Sets & Number Theory
1.a). Pierre was given 4 sets as follows by his teacher
P = {w,x,y,z}
Q = {11, 14, 15, 16}
R = {a,b,c,d,x,y,z}
S = {4,6,8}
Help him in finding the answers for the following.
i) n(R)=
ii) what is the number of subsets of S and also write any 4 subsets of it.
b) Pierre wanted to represent the information of sets P and R given above in the Venn Diagram
given below .Copy and complete the given Venn diagram.

P
R
c) 46 tourists took a holiday together in Switzerland and Denmark. 30 had already visited
Switzerland and 28 had already visited Denmark but 4 visited neither of the place.
Calculate
.
i) The number of tourists who visited both Switzerland and Denmark and also shade the
region in the Venn diagram which represents this information by drawing a separate
venn diagram in your answer sheet.
ii) the number of tourists who visited only Switzerland
iii) the number of tourists who visited Denmark but not Switzerland
Pierre expresses his thanks to you for helping him in his homework.
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Page 4 of 9
IGCSE MENTORS
Sets & Number Theory
Sets
Grade : 9
IGCSE MATHEMATICS
CAMBRIDGE
Sets :
A set is a well defined group of object or symbol. The object or symbol are called the element of the
set .
Write the another two element of the set in the following :1) { Asia , Africa , Europe , ………….. }
2) { 2 , 4 , 6 , …………………. }
3) { Sunday , Monday , Tuesday ………….. }
4) { 1, 3, 6, 10 ………………….}
5) { a, e, i …………….. }
6) { Earth, Mars, venus, …………………}
Subsets: If all the element of one set X are also the element of another set Y, then X is said to be a subset of Y
This is written as X .
1) A = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(iii)
List subset B ( even numbers )
(iv)
List subset C ( prime numbers )
2) P = whole number less than 50
a) List the subset Q ( even numbers )
b) List the subset R ( odd numbers )
c) List the subset S ( prime numbers )
d) List the subset T( square numbers )
3) P = whole number between 60 and 100
a) List the subset B ( multiples of 5 )
b) List the subset C ( multiples of 3 )
c) List the subset D( Square numbers )
Venn diagrame
Page 5 of 9
IGCSE MENTORS
Sets & Number Theory
1) If A = a b c d e f g h i j k l m
B = cdelmnopqrs
Represent on a venn diagrame
A
B
n
a b f cd
g h
e l
i j k m
o
p
q
r
s
2) If  = 1 2 3 4 5 6 7 8 9 10 11 12
A = 1 2 5 6 11 12
B = 7 8 9 10 11 12
Represent on a venn diagrame

A
1
B
2
11
7
5
8
9
6
12
10
3
4
3. Answer the following with the help of venn diagrame
(i) A U B =
(ii) A ∩ B =
(iii) A' U B =
(iv) A' ∩ B =
(v) A' U B' =
(vi) A' ∩ B' =
(vii) ( AUB )' =
(viii) ( A∩B )' =
A

1
B
2
7
8
3
9
14
10
4
11
5
12
6
13
15
16
Page 6 of 9
IGCSE MENTORS
Sets & Number Theory
4. In a class of 35 students, 19take Spanish, 18 take French and 3 take neither . calculate how
many take :a) both French and Spanish
b) just Spanish
c) just French
5. 46 tourists took a holiday together in Switzerland and Denmark. 30 had already visited
Switzerland and 28 had already visited Denmark but 4 visited neither of the place.
Calculate
i. The number of tourists who visited both Switzerland and Denmark
ii. the number of tourists who visited only Switzerland
iii. the number of tourists who visited Denmark but not Switzerland
6. In a class of 30 students, 14 take physics, 16 take chemistry and 4 take neither.
a) Enter the above information in a Venn diagram
b) Form an equation in x
c) Find the value of x
d) Hence find how many takes only physics
7. In a class of 33 students, 19 take Dhivehi, 16 take Hindi and 4 take neither.
d. Represent the above information on a Venn-diagram.
e. Calculate how many take both Dhivehi and Hindi.
f. Calculate how many take only Dhivehi.
8. In a class of 30 students, 14 take physics, 16 take chemistry and 4 take neither.
a) Enter the above information in a Venn diagram
b) Form an equation in x
c) Find the value of x
d) Hence find how many takes only physics
e) Shade the region (P U C1)
9. a). The set A and set B are such that n(A) = 28, n(B)=12 and n(AB) = 36,
find n(AB).
(b) State Trading company has 43 employees and they drink either Tea or Coffee
or both. If 28 of them drink Tea, 10 of them drink Tea but not coffee.
(i) By drawing a Venn diagram, find the number of employees who drink
both Tea and Coffee.
(ii) Find the number of employees who drink Coffee but not Tea.
(iii) Write the region ‘d’ shown below, using set notation.
T
a=
C
b=
d=
Page 7 of 9
IGCSE MENTORS
Sets & Number Theory
10. Complete the following statement.
ξ
A
B
8
1
4
2
7
3
9
6
5
{…….}
A’ = {…….}
a) ξ =
b)
n( A∩B) = ……..
11.
The given Venn diagram shows the number of people visited the carnival on the
first day (F) and on the second day (S)
S
F
x
(x-100)
(x – 200)
i) Write down an equation in x if a total of 600 people visited in two days
ii) Solve the equation to find the value of x
iii) Hence find how many visited the carnival both the days
iv) Find n (S).
SHADING
Shade the region as mentioned below and copy the figure for each of the question.
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv)
(xvi)
AUB
A∩B
A' U B
A ∩B'
(AUB)'
(A∩B)'
A' U B'
A' ∩ B'
A
B
################################################################################
***************************************************
Page 8 of 9
IGCSE MENTORS
Sets & Number Theory
NUMBER THEORY
Natural numbers N = { 1, 2, 3, 4, 5……………. }
Whole numbers W = { 0, 1, 2, 3, 4,……………..}
Integers
Z = { ……., -2, -1, 0, 1, 2, ……….}
Positive integers Z+ = { 1, 2, 3, 4, ……………..}
Negative integers Z- = { -1, -2, -3,………………..}
Odd numbers = { 1, 3, 5, 7……………………}
Even numbers = { 2, 4, 6, 8 ……………….. }
Prime numbers = { 2, 3, 5, 7, 11, ……………….}
Perfect squares = { 1,4,9,16,25………………..}
Rational numbers : Number that can be written as a fraction { in the form a / b , where a , b are
integers , b ≠ 0 }
Ex:- 0.7, 22/7, 17/14, cos60
Irrational numbers : numbers that can not be written as a fraction { cannot be expressed as a / b
form }
Ex : √2 , Π, √12
Question 1
25 , cos60o, 14, 0.7,
3
, 12 , ∏
8
a) write down any two rational numbers.
A number is selected at random from the above set of values.
Find the probability that the number is
b) an integer.
c) an irrational.
Question 2
1

2

R = {Rational numbers}
I = {Integers}


  7, 2 , ,  , 81,1,0,2.5
a) Place the 8 numbers in the correct places on the Venn diagram.
R
ξ
I
Page 9 of 9
IGCSE MENTORS