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Sets
A set is a well defined group of objects such as those shown below.
A set can be finite or infinite. The members of the set are called
elements “e”. If an element e belongs to a set S then we write e  S
(e is an element of S).
If an element e does not belong to a set S then we write e  S (e is
not an element of S).
European Capitals
A ={London, Rome, Paris, Cairo,…}
Rome  A
Munich  A
Letters of the Alphabet
C ={A, B, C, D, E,……Z}
QC
βC
B ={3,7,9,14}
7B
20  B
Square Numbers
D ={1,4,9,16,25,36,…..}
100  D
63  D
Sets
Describe in words the sets below and state whether they are finite or
infinite. Where possible, use set notation to write down one additional
element that is a member of the set and one that isn’t.
A ={red, orange, yellow, green, blue, indigo, violet} Colours of the rainbow
B ={1,3,6,10,…} Triangular numbers (infinite)
C ={Rebecca, Razia, Rose, Rhea, Razaanah, …} Girls’ names beginning with R.
D ={August, January, April, …} Months of the year
E ={Monday, Tuesday, Wednesday, …} Days of the week
F ={64, 27, 8, 125, …} Cube numbers (infinite)
G ={2,3,5,7,11,13, …} Prime numbers (infinite)
Sets
Subsets
If all the elements of a set A are also elements of another set B then
A is said to be a subset of B (A  B).
Write down some sets that
A ={August, January, April, …}
are subsets of others from
the list below.
B ={Monday, Tuesday, Wednesday, …}
C ={64, 27, 8, 125, …}
D ={2,4,6,8,10,12, …}
E ={2,3,5,7,11,13, …}
F ={1,2,3,4,5,6, …}
G ={September,October,November}
H ={Saturday,Sunday}
I ={1,343,1000}
GA
IF
HB
CF
IC
DF
E F
Sets
Subsets
Notation:  (Subset),  (Proper Subset)
 (Not a subset)
 (The empty Set)
 (Not a proper subset)
Consider the set A consisting of three people, Alice, Stephen and Jane.
A ={Alice, Stephen, Jane}
We can list all subsets of A as follows:
B = {Alice, Stephen, Jane} B  A
C = {Alice, Stephen} C  A
D = {Alice, Jane} D  A
E = {Stephen, Jane} E  A
F = {Alice} F  A
G = {Stephen} G  A
H = {Jane} H  A
I=
BA
By convention, sets B
and I are considered
as subsets of A.
We distinguish these
from proper subsets
by use of  symbol.
We can write F  A or G  D
Sets
Subsets
Notation:  (Subset),  (Proper Subset)
 (Not a subset)
 (The empty Set)
 (Not a proper subset)
Question 1. If Set A are the whole numbers less than 20, list:
(a) The subset B {odd numbers}, (b) The subset C {prime numbers},
(c) The subset D {Multiples of 5).
(a) B = {1,3,5,7,9,11,13,15,17,19}, (b) C = {2,3,5,7,11,13,17,19}, (c) D = {5,10,15).
Question 2. List all subsets of {STU} indicating which are proper/improper subsets.
{STU} and  are improper subsets. {S,T},{S,U},{T,U),{S},{T},{U} are proper subsets.
Question 3. State whether the following statements are true or false.
(a) {4,5,6,7}  {4,5,6,7}
False
(c) {1,2,3}  {prime numbers}
True
(b) {Red, Yellow, Blue}  {Yellow, Blue, Red}
True
(d) {A,B,C,D,E}  {B,E,D,F,E,A}
False
Sets
The number of elements of a set S can be written as n(S).
List the members of the given sets below and state n(S) in each case.
A ={Odd numbers < 12}
1,3,5,7,9,11
n(A) = 6
B ={Even numbers  14} 2,4,6,8,10,12,14
n(B) = 7
C ={Square numbers between 20 and 70} 25,36,49,64
D ={Vowels} a,e,i,o,u
n(C) = 4
n(D) = 5
E ={Months with 30 days} September, April, June, November n(E) = 4
F ={prime numbers p; 11 p 39} 11,13,17,19,23,29,31,37
G ={Cube numbers C: 10 C 50}
27
n(G) = 1
n(F) = 8
Sets
Intersections and Unions
Consider the sets A and B below where 9 and 14 are elements common to
both sets. We write A  B = {9,14} (A intersection B).
The union of both sets is all elements that are contained in A or B or
both. We write A  B = {3,7,9,14,28} (A union B).
A ={3,7,9,14}, B = {9,14,28}, C={5,7,12,14,24}
Write the following sets:
P=A C
P ={7,14}
Q=B C
Q ={5,7,9,12,14,24,28}
R=B C
R ={14}
S=A C
S ={3,5,7,9,12,14,24}
Sets
The Universal Set and Compliments
The universal set is the set that contains all elements under consideration
for a particular problem. This set can be denoted by the letter E.
Consider the universal set shown together with some subsets.
E ={1,2,3,4,5,6,7,8,9,10}
A ={3,7,9,10}, B = {1,2,3}, C={2,4,6,8,10}
We define the compliment of a set A as the set of elements A’ that are
in E but not in A.
So we can write A’ ={1,2,4,5,6,8}
Write out B’ and C’
B’ ={4,5,6,7,8,9,10}
C’ ={1,3,5,7,9}
Sets
The Universal Set and Compliments
Question:
(a) Give a suitable universal set E for the subsets below.
(b) Find A  B
(c) A  C
(d) B’
A ={July, August September}
B ={Months with less than 31days}
C ={Months with four-letter names}
(a) E ={Months of the year}
(b) {September}
(c) {June, July, August, September}
(d) {January, March, May, July, August, October, December}
Sets
The Universal Set and Compliments
For questions 1 to 4 below find: (a) A  B
1. A = {1,2,3},
2. A = {1,3,5,7},
(b) A  B
B = {2,3,5,7}
B = {2,3,5,8}
3. A = {Multiples of 4 less than 20}, B = {Multiples of 6 less than 20}
4. A = {Square numbers below 100}, B = {Cube numbers below 100}
1 (a) {2,3},
(b) {1,2,3,5,7}
2 (a) {3,5},
(b) {1,2,3,5,7,8}
3 (a) {12}, (b) {4,6,8,12,16,18}
4 (a) {1,64}, (b) {1,4,8,9,16,25,27,36,49,64,81}
Sets
Venn Diagrams
Venn diagrams are a way of showing sets pictorially in diagrammatic form.
The universal set is represented by a rectangle and subsets are shown in
circles within the rectangle. When drawing Venn diagrams we must ensure
that the number of intersections fit the given data.
Example 1
If E = {1,2,3,4,5,6,7,8,9,10}
A = {1,3,5,7,9,10}
AB
E
A
B
1
9
10
2
B = {3,4,5,7,8}
3
5
7
4
6
8
Venn Diagrams
Sets
Venn Diagrams
Venn diagrams are a way of showing sets pictorially in diagrammatic form.
The universal set is represented by a rectangle and subsets are shown in
circles within the rectangle. When drawing Venn diagrams we must ensure
that the number of intersections fit the given data.
Example 2
If E = {1,2,3,4,5,6,7,8,9,10}
A = {2,4,6,8,10}
B = {4,6,8}
E
A
B
1
3
2
5
4
6
8
7
10
9
Sets
Venn Diagrams
Venn diagrams are a way of showing sets pictorially in diagrammatic form.
The universal set is represented by a rectangle and subsets are shown in
circles within the rectangle. When drawing Venn diagrams we must ensure
that the number of intersections fit the given data.
Example 3
If E = {1,2,3,4,5,6,7,8,9,10}
A = {2,4,6,8,10}
B = {1,3,5}
E
A
2
4
7
B
1
6
8
10
3
5
9
Sets
Venn Diagrams
Venn diagrams are a way of showing sets pictorially in diagrammatic form.
The universal set is represented by a rectangle and subsets are shown in
circles within the rectangle. When drawing Venn diagrams we must ensure
that the number of intersections fit the given data.
Example 4
If E = {1,2,3,4,5,6,7,8,9,10} A = {2,4,5,8,10} B = {4,6,9,10}
C = {1,4,6,}
E
A
2
5
8
3
B
10
4
1
C
9
AB C
6
7
Sets
Venn Diagrams
Show each group of sets in a Venn diagram.
(a) E = {1,2,3,4,5,6,7}
A = {2,3,4,5}
B = {2,5}
(b) E = {1,2,3,4,5,6,7}
A = {1,2,6}
B = {2,4,6}
(c) E = {1,2,3,4,5,6,7}
A = {1,2,4}
B = {3,6}
(d) E = {1,2,3,4,5,6,7,8} A = {2,4,5,8} B = {1,2,7,8,}
C = {2,4,6,}
Sets
Venn Diagrams
(a) Describe in words the elements of: (i) Set A (ii) Set B (iii) Set C
(b) Copy and complete the following statements:
(i) A  B = {…} (ii) A  C = {…} (iii) B  C = {…} (iv) A  B  C = {…}
(v) A  B = {…} (vi) C  B = {…}
B
A
5
12
15
11
18
3
7
13
17
C
6
9
19
(a) (i) Odd numbers from 3 to 15
(ii) Multiples of 3 from 3 to 18
(iii) Some odd primes  19
(b)
(i) A  B = {3,9,15}
(ii) A  C = {3,7,11,13}
(iii) B  C = {3}
(iv) A  B  C = {3}
(v) A  B = {3,5,6,7,9,11,12,13,15,18}
(vi) C  B = {3,6,7,9,11,12,13,15,17,18,19}
Sets
In each of the Venn diagrams below, describe the shaded area in terms of the
E
E
E
subsets.
(a)
(b)
(c) A
B
B
A
B
A
(A  B)
(d)
A
(A  B)
E
(e)
B
(f)
A
(A  B  C)
B
A
B
E
(h)
B
B’
A
(A  B’ 
A’ or (A  B)’
E
B
C
E
A
B
C
(g)
B’
E
(i)
A
B
C
(A  B  C’)
E
Sets
Example Question: In the Venn diagram below E is the number of pupils in a
year 7 class that attend an after school sporting activity at a local gym.
A = {students who play squash}, B = {students who play volley ball}
(a) How many students are in this year 7 class?
(b) How many students play squash?
(c) How many students play both squash and volley ball?
(d) How many students play neither squash nor volley ball?
E
B
A
3
8
7
(a) n(E) = 3 + 8 + 7 + 12 = 30
(b) n(A) = 3 + 8 = 11
(c) n(A  B) = 8
12
(d) n(A  B)’ = 12
Sets
Question: In the Venn diagram below E is the number of people that attended
a local council meeting.
A = {people that voted}, B = {people that asked for tea}
(a) How many people asked for tea?
(b) How many people asked for tea and voted?
(c) How many people neither asked for tea nor voted?
(d) How many people attended the meeting?
E
B
A
36
30
23
(a) n(B) = 23 + 30 = 53
(b) n(A  B) = 30
(c) n(A  B)’ = 20
20
(d) n(E) = 36 + 30 + 23 + 20 = 109
Sets
PROBLEM SOLVING
Example Question: In a class of 30 students, some studied physics and some
chemistry. If 20 studied physics, 18 studied chemistry and 4 studied neither,
calculate the number of students that studied both subjects.
E
P
20 – x + x + 18 – x + 4 = 30
C
8
20 - x
12
x
42 - x = 30
6
18 - x
x = 12
4
Sets
PROBLEM SOLVING
Question: In a sports survey, 324 teenagers responded to a questionnaire.
They were asked if they liked football and rugby. 180 said they liked football,
159 said they liked rugby and 30 said they liked neither. Calculate how many
teenagers:
(a) liked both football and rugby. 45
(b) Liked only rugby. 114
E
F
R
135
180 - x
45
x
114
159 - x
30
180 – x + x + 159 – x + 30 = 324
369 - x = 324
x = 45