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Sets
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
2
Algebra
Chapter 2
Sets
2.1
Introduction
2
2.2
Venn Diagrams
3
2.3
Equality of Sets
3
2.4
Subsets
4
2.5
Empty Set and Singleton
5
2.6
Operations on Sets
5
2.7
Number of Elements in a Finite Set
10
Prepared by K. F. Ngai
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1
Sets
Advanced Level Pure Mathematics
2.1
Introduction
1.
A set is a collection of definite, distinguishable objects.
e.g.
{ chairs , desks , tables } is a set of furniture.
{ a , e , i , o , u } is a set of vowels.
{ 0 , 1 , 1 , 2 , 2 , 3 , 3 , ... } is a set of integers.
2.
The objects of a set is called the elements or members of the set.
e.g.
3.
Expressing a set in two ways:
(a)
(b)
4.
a , e , i , o and u are the elements of the set of vowels.
Tabular form :
e.g.
{ a , e , i , o , u } is the set of vowels.
e.g.
{0 , 2 , 2 , 4 , 4 , ...} is a set containing all the even numbers.
Set-Builder form :
e.g.
{ x : x is a vowels } is the set of vowels.
e.g.
{ x : x is an even number } is a set containing all the even numbers.
If x is a member of a set A , we write x  A.
If x is not belong to a set A , we write x  A.
5.
e.g.
If A = { x : x is a vowel } , then a  A , u  A , but g  A.
e.g.
If B = { x : x is an integer greater than 11 } , then 20  B , 999  B ,
but 11  B.
Sets containing finite number of objects are called finite sets. Otherwise, it is
called an infinite set.
e.g.
{ x : x is a vowel } is a finite set.
e.g.
{ x : x is an integer greater than 11 } is a infinite set.
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Sets
Advanced Level Pure Mathematics
6.
Some symbols are frequently adopted:
1.
N denoted the set of all positive integers.
N = { 1, 2, 3, 4, ... }
2.
Z denoted the set of all integers.
Z = { 0, 1, 1, 2, 2, ... }
2.2
p
: p, q  Z and q  0 } is the set of rational numbers.
q
3.
Q={
4.
Q = { x : x  Q and x > 0 }
5.
R = { x : x is a real number } is the set of real numbers
6.
R = { x : x  R and x > 0 }
7.
C = { x + yi : x , y  R and i2 = 1 } is the set of complex numbers
Venn Diagrams
A = {a, e, i, o, u} , B = {1, 2, 3, 4} , C = {3, 4, 5, 6, 7} , D = {5}
A
D
u
B
a
5
e
i
3
o
1
7
4
6
C
2
2.3
Equality of Sets
Def. Two sets are said to be equal if and only if they contain the same elements.
i.e.
A=B
iff
xA
e.g.
{1, 1, 2, 3, 3, 3, 3} = {1, 2, 3}
e.g.
{1, 2, 3, 4, 5} = {2, 5, 4, 1, 3}

xB
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Sets
Advanced Level Pure Mathematics
2.4
Subsets
Def.2.1
A set A is said to be a subset of another set B , denoted by A  B,
if and only if every element of A is an element of B.
i.e. A  B
e.g.
{1, 2, 3}  {1, 2, 3, 4, 5}
e.g.
{4, 5, 6}  {1, 2, 3, 4, 5}
e.g.
ZQR.
Def.2.2
x, xA
iff

xB
A
B
If A  B and A  B , A is called a proper subset of B.
(a) A  A , for every set A.
Thm.2.1
(b) if A  B and B  C , then A  C.
(c) A = B
Proof :
A  B and B  A.
if and only if
(a)
x, xA
AA.

(b)
Since A  B
x, xA
xA

x  B ............(1)
Also , since B  C
x, xB 
x  C .............(2)
From (1) and (2), we have
x, xA  xC
i.e. A  C
N.B. A  B
iff
 x , x  A and x  B.
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Sets
Advanced Level Pure Mathematics
2.5
Empty Set and Singleton
Def.2.3
The empty set, denoted by , is a set which contains no element.
 = { x : x  x } is unique
i.e.
e.g.
{x|
x > 3 and x < 3 } is an empty set
e.g.
{x|
x is an real number and sin x = 2 } is an empty set
Thm.2.2
For any set A ,   A
N.B.   N  Z  Q  R  C
Def.2.4
e.g.
A set containing exactly one element is called a singleton.
{x}, {2}, {} is a singleton

N.B. (1)
(2)
2.6
{} is not an empty set
  {}
Operations on Sets
Def.2.5
The intersection of A and B , denoted by A  B , is defined as
A  B = { x | x  A and x  B }.
i.e.
A
xAB
AB
iff
x  A and x  B
B
A
B
AB
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Sets
Advanced Level Pure Mathematics
e.g.
Let A = { a, b, c, d } , B = { c, d, e, f } . Then
e.g.
Let A = { x  Z : x  3 }, B = { x  Z : x < 8 } .
A  B = {c, d }
Then A  B = { x  Z : 3  x < 8 }
e.g.
A = { x  Z : x is an even number } , B = { x  Z : x is the multiple of 3 }
Then A  B =
Def.2.6.
Two sets A and B are said to be disjoint iff A  B = .
A
e.g.
B
A
B
Let A = {1, 2, 3} and B = {4, 5}. Are A and B disjoint?
Soln: Since A  B =  , A and B are disjoint.
Thm. If C is a subset of the sets A and B , then C  A  B.
Def.2.7
The union of A and B , denoted by A  B , is defined as
A  B = { x | x  A or x  B }.
i.e.
xAB
iff
x  A or x  B
AB
A
e.g.
AB
B
A
B
Given that A = {1, 2, 3, 4, 5} , B = {3, 4, 5, 6, 7} , C = {1, 3, 5, 7}.
Then
AB=
ABC=
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Sets
Advanced Level Pure Mathematics
Thm.2.3
Let A , B and C be any three sets. Then we have
(a)
(b)
(c)
(d)
(e)
Thm.2.4
Thm.2.5
Idemptotent Laws:
(i)
AA=A
(ii)
AA=A
Commutative Laws:
(i)
AA=AA
(ii)
AA=AA
Associative Laws:
(i)
(A  B)  C = A  (B  C)
(ii)
(A  B)  C = A  (B  C)
Distributive Laws:
(i)
(A  B)  C = (A  C)  (B  C)
(ii)
(A  B)  C = (A  C)  (B  C)
(iii)
C  (A  B) = (C  A)  (C  B)
(iv)
C  (A  B) = (C  A)  (C  B)
(v)
(A  B)  (C  D) =
Identity Laws:
(i)
A=
(ii)
A   = A.
Let A and B be any two sets. Then
(a)
ABA
and
ABB
(b)
AAB
and
B  A  B.
Let A , B , C and D be any four sets. If A  C and B  D , then
(a)
ABCD
(b)
A  B  C  D.
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Sets
Advanced Level Pure Mathematics
Def.2.8
Let A be a subset of E. The complement of A in E is the set
A = E \ A = { x : x  E and x  A }.
E
E \ A = A
where
AE
A
e.g.
If
A = { x : x  3 or x  8 } , then
R\A={x:3<x<8}
e.g. R \  =
, R\R=
Def.2.9
The relative complement of a set B in another set A is the set
A \ B = { x : x  A and x  B } = A  B
E
B \ A = B  A
A\B
A
e.g.
B
Let A = { 1, 2, 3, 4, 5 },
Then
B = { 2, 4, 6, 8 },
A\B=
C = { 1, 3, 5 }
B\A=
C\B=
Thm.2.6
where B = E \ B
C\A=
Complement Laws:
Let A be a subset of a set E. Then
Proof :
(a)
A\A=
(e)
A  A =
(b)
A\=A
(f)
A  A =
(c)
\A=
(g)
E =
(d)
(A) = A
(h)
 =
(a)
xA\A
Hence

xA

x
and
xA
A\A=
Alternatively,
A \ A = A  A = A  (E \ A) = 
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Sets
Advanced Level Pure Mathematics
(d)
Theorem
x  (A)

xE
and
x  A

xE
and
xA

xA
Let A and B be a subset of a set E. Then
(A \ B) = A  B'
e.g.1 Let A and B be two subsets of a set E . Prove that
(a)
(A \ B)  (B \ A) = 
(b)
A  (B \ A) = 
Soln.: (a)
(A \ B)  (B \ A) =
=
=
=
(A  B)  (B  A)
(A  A)  (B  B)


e.g.2 Let A and B be any two sets.
Prove that if A \ B =  and B \ A =  , then A = B.
Thm.2.7
De Morgan‘s Laws:
Let A and B be two subsets of E. Then
(a)
(A  B) = A  B
(b)
(A  B) = A  B,
where
A = E \ A,
B = E \ B.
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Sets
Advanced Level Pure Mathematics
2.7
Number of Elements in a Finite Set
1.
Let n(A) denote the number of elements in a set A.
e.g.
If A = {a, b, c, d}, then n(A) = 4
2.
n(A  B) = n(A) + n(B)  n(A  B)
If A  B =  , n(A B) = n(A) + n(B)
3.
n(A \ B) = n(A)  n(A  B)
e.g.6 In a class of 40 students, every student studies either Biology or
Geography.
If there are 30 students studying Biology and 20 students studying Geography,
how many students study both?
Soln : Let
A = the set of students studying Biology
B = the set of students studying Geography
Then A  B = the set of students studying both
n(A  B) = n(A) + n(B)  n(A  B)
= 30 + 20  40
= 10
Hence, 10 students study both subjects.
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