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1. SET Theory-1
Introduction to mathematics (University of Zambia)
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1. SET THEORY
We find the concept of a set in many branches of mathematics. In this chapter, we will
discuss the theory of sets and later different sets of numbers.
1.1 BASIC DEFINITIONS
Definition 1.1.1
A set is a collection of objects. The objects in the set are called elements of the set.
A set can be given by description or by listing the elements. We usually indicate a set by a
pair of braces   but parentheses   and square brackets   are sometimes used as well.
Example 1.1.1
The following are sets:
(a) Top 10 selling fruits in Zambia.
(b) 1, 2,9,10
(c) {x : x  2, x  }  [2, )
(d) { y 
: 0  y  1}  (0,1)
Definition 1.1.2
Two sets are equal if they contain exactly the same elements.
Example 1.1.2
1. The set {1,3, 4,5} is equal to the set {1, 4,3,5}.
2. The set containing the letters of the word “SILENT” is equal to the set containing the
letters of the word “LISTEN”.
NOTATION
We usually use a lower case letter, say x to represent a generic element of a set and the
symbol "  " to mean “is an element of” or “is a member of”. For example, x  A would be
read as " x is an element of a set A " or " x is a member of a set A". If a set A is equal to a
set B , we write A  B.
Definition 1.1.3
A universal set, denoted by U or E , is defined as the set containing all possible elements of
any set we wish to consider.
Example 1.1.3
What would be the universal set for the sets in Example 1.1.1?
(a) Fruits
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(b) Whole numbers or Integers or Real numbers.
Definition 1.1.4
A set with no elements is called the empty set and is denoted by {} or  .
Definition 1.1.5
A is a subset of B if every element of A is also an element of B.
NOTATION
If A is a subset of B and can also be equal to B , we denote this by A  B. If A is a subset
of B and A  B, then A is said to be a proper subset of B and we use the notation A  B.
NOTE: If A  B and B  A, then A  B.
Sets can also be represented pictorially in a diagram called a Venn diagram. In a Venn
diagram, a rectangle is drawn to represent the universal set and closed figures (usually
circles) are drawn inside the rectangle to represent other sets.
U
B
A
C
Note from the Venn diagram above that
1. every set is a subset of the universal set U .
2. A  B.
1.2 SET OPERATIONS
There are basically three set operations. These are the union, the intersection and the
complement of sets.
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Definition 1.2.1
If A and B are sets, then the union of A and B , denoted by A  B, is the set containing all
elements of both A and B , i.e.
A  B  {x : x  A or x  B}.
Definition 1.2.2
The intersection of two sets A and B , denoted by A  B, is the set containing all elements
that are found in both A and B , i.e.
A  B  {x : x  A and x  B}.
Definition 1.2.3
The complement of a set A, denoted by Ac or A, is the set of all elements that are not in A,
i.e.
Ac  {x : x U , x  A}.
Definition 1.2.4
The difference of two sets A and B , denoted by A  B, is the set of elements which belong
to A and which do not belong to B , i.e.
A  B  {x : x  A and x  B c }
 A  B  A  Bc .
Example 1.2.1
Given that E  {2, 1, 0,1, 2,3, 4,5, 6}, A  {2,1,3}, B  {0,1, 2,3} and C  {2, 4}. Find
(a) A  B (b) B  C (c) A  B  C (d) A  B (e) B  C (f) A  B  C (g) A  C


(h) A  B  C (i) A   B  C  (j) Ac  Bc  C .
c
Represent the sets on a Venn diagram.
Solutions
(a) A  B  {2, 0,1, 2,3}
(b) Exercise
(c) A  B  C  {2, 0,1, 2,3, 4}
(d) Exercise
(e) B  C  {2}
(f) Since B  C  {2} , we have that A  B  C   .
(g) Exercise
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(h) A  B  C  A   B  C  .
c
Since B  C  B  C c  {0,1,3}, we have that  B  C   {2, 1, 2, 4,5,6} . Thus,
c
A  B  C  A   B  C   {2}.
c
(i) Since  B  C   {2, 1,0,1,3, 4,5,6}, we have that A   B  C   {2,1,3}  A.
c
c
(j) Ac  {1,0, 2, 4,5,6}, B c  {2, 1, 4,5, 6} and Bc  C  {4}
 Ac   Bc  C   {1,0, 2, 4,5,6}  Ac
U
B
A
-2
1
0
3
2
C
4
-1
5
6
1.3 LAWS OF ALGEBRA OF SETS
Sets under the above operation satisfy various laws listed below:
1. The idempotent laws:
(a) A  A  A
(b) A  A  A
2. The Associative laws:
(a) A  ( B  C )  ( A  B)  C
(b) A  ( B  C )  ( A  B)  C
3. The Commutative laws:
(a) A  B  B  A
(b) A  B  B  A
4. The Distributive laws:
(a) A  ( B  C )  ( A  B)  ( A  C )
(b) A  ( B  C )  ( A  B)  ( A  C )
5. The Identity laws:
(a) A    A
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(b) A    
(c) A  E  E
(d) A  E  A
6. The Complement laws:
(a) A  Ac  E
(b) A  Ac  
(c) ( Ac )c  A
(d) E c  
(e)  c  E
7. De Morgan’s laws:
(a) ( A  B)c  Ac  B c
(b) ( A  B)c  Ac  B c
Example 1.3.1
Simplify each of the following:
(i)  A  ( A  B) (ii) A  ( Ac  B)
Solutions
We use laws of algebra of sets.
(i)  A  ( A  B)  ( A)  ( A  B), by De Morgan’s laws
 A  ( A  B), by Complement laws
 A  ( A  ( B)), by De Morgan’s laws
 ( A  A)  ( A  B), by Distributive laws
   ( A  B)
 A  B, by Identity laws
(ii) A  ( Ac  B)  A  ( Ac  B c )c
 A  ( Ac )c  ( B c )c 
 A  ( A  B)
 ( A  A)  ( A  B )
 A  ( A  B)
 A.
1.4 SETS OF NUMBERS
Definition 1.4.1
Natural numbers are numbers used for counting, i.e. 1, 2,3,....
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We denote natural numbers by
.
Definition 1.4.2
Integer numbers (Integers) are natural numbers with their opposites (natural numbers with a
negative sign), and 0, i.e. ..., 2, 1, 0,1, 2,...
The notation for integers is
.
NOTE:
1.
can also be referred to as positive integers.
2.
can further be divided into odd and even numbers.
Definition 1.4.3
Rational numbers are numbers that can be expressed in the form
a
, where a and b are
b
integers and b  0.
Rational numbers are denoted by
NOTE:
and
are also
.
.
The opposite of rational numbers are irrational numbers (denoted by IR ) . For example,
numbers such as 2, 3 and e are irrational numbers because they cannot be expressed as
ratios of integers.
Definition 1.4.4
Real numbers are a collection of rational and irrational numbers and are denoted by
.
In others words, any number on the continuous line is a real number.


Definition 1.4.5
Complex numbers, denoted by
, are numbers that can be expressed as z  x  iy, where
x, y  and i  1. x is referred to as the real part while y is referred to as the imaginary
part of a complex number, i.e.
Re( Z )  x and Im( Z )  y.
NOTE:
are also complex numbers with the imaginary part equal to zero.
IR
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Example 1.4.1
Classify the following numbers:
0.3333..., e 2 ,  9,
2
, 2  i, 3 1   i  ,1000.
3
Solutions
ALL
0.3333...,  e 2 ,  9,
e 2 ,
IR
2
,1000
3
2
3
0.3333...,  9, 1000
 9, 1000
1000
1.4.1 Real Numbers
The set of real numbers can be expressed in form of an interval, i.e.
can be expressed as follows:
subsets of
1. (a, b)  {x 
2. (a, b]  {x 
3. [a, b)  {x 
4. [a, b]  {x 
5. (, a)  {x 
: a  x  b}
a
b
a
b
a
b
a
b
: a  x  b}
: a  x  b}
: a  x  b}
: x  a}
a
6. (, a]  {x 
: x  a}
a
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 (, ). Other
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7. (a, )  {x 
: x  a}
a
8. [a, )  {x 
: x  a}
a
We can do set operations of sets involving intervals as the next example shows:
Example 1.4.2
Given that the universal set is
and that A  (1, 7], B  [5,3] and C  [1,10], find each
of the following sets and display on the number line:
(i) Ac (ii) B  A
(iii) ( B  A)  C 
Solutions
(i)
Ac
Ac
A
1
0
7
 Ac  (, 1]  (7, )
(ii) B  A  B  Ac
B
Ac
Ac
5
1 0
3
7
 B  A  B  Ac  [5, 1]
(iii)
B
B A
A
C
C
C
5
1 0
3
7 10
B  A  (1,3] and C   (, 1)  (10, ). Therefore, ( B  A)  C    .
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1.4.2 Rational Numbers
Any rational numbers can be represented as a decimal and for some, numbers repeat after a
decimal point. Given any rational number in form of a decimalwith some figures repeating
a
after a decimal point, we can express it in the form , a, b  , b  0. We place a bar over
b
the set of numbers which is to be repeated indefinitely.
Example 1.4.3
Express the following rational numbers in the form
(i) 3.14
(ii) 1.142857
a
, a, b  , b  0 in their lowest terms:
b
(iii) 0.29432
Solutions
(i) Let x  3.14. Multiplying throughout by 100, we get 100x  314.14. We can now subtract
the two numbers. We need to make sure that numbers after the decimal point are the same so
that after subtraction we get zero after the decimal point. Thus,
100 x  314.14
 x 
3.14
99 x  311.0
x
311
311
which is already in its lowest terms. Therefore, 3.14 
.
99
99
(ii) Exercise
(iii) Let x  0.29432. Multiplying throughout by 100, we get
100 x  29.432
(i)
We now multiply by 1000 to get the same repeating numbers after the decimal point.
100000 x  29432.432
(ii)
Subtracting (i) from (ii), we get
100000 x  29432.432
 100 x 
29.432
99900 x  29403
x 
29403 1089
, i.e.

99900 3700
0.29432 
1089
.
3700
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1.4.3 Irrational Numbers
We can prove that a number, say
2 is irrational.
Example 1.4.4
Prove that
2 is irrational.
Solutions
We prove by contardiction. Suppose that
2 is rational, i.e.
a
, a, b  , b  0
b
2
And that a and b have no common factor. Then,
2
a

b
 
2
2
a
 
b
2
a2
b2
 a 2  2b 2 .
2
Thus, a 2 is divisible by 2 implying that a is also divisible by 2. Since a is divisible by 2,
it can be written as a  2k , k  . Then,
(2k ) 2  2b 2
 4k 2  2b 2
 b 2  2k 2 .
Thus, b 2 is divisible by 2 implying that b is also divisible by 2. This means that a and b
have a common factor 2. This is a contradiction. Hence, 2 is irrational.
1.4.4 Surds and Rationalisation of the Denomenator
The square root of a prime number is a surd, for example,
2 , 3 and
PROPERTY
If a and b are real numbers, then
(a)
ab  a . b
(b)
a
a
 
b
b
(c)
a a a
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Example 1.4.5
Simplify each of the following:
20  2 45  80
(a)
(b)


11  5 5  11

(c)
12
156
Solutions
20  2 45  80  4  5  2 9  5  16  5
(a)
 4. 5  2. 9. 5  16. 5
 2 5  2.3 5  4 5
 2 56 54 5
4 5
(b)



11  5 5  11  5 11  11. 11  25  5 11
 10 11  11  25
 10 11  36
12
4 3
2 3
1



156
4 39 2 3 13
13
(c)
If a fraction has a number with surds in the denominator, then we can ‘remove’ the surd from
the denominator. We can do this by multiplying by a special ‘1’. When this is done, we say
that the denominator has been rationalised.
Example 1.4.6
Rationalise the denominator of each of the following:
(a)
1
2
(b)
1 2
1 2
(c)
20  5  18
1  48
Solutions
(a)
(b)
(c)
1
1
2
2



.
2
2
2
2






1 2 1 2
1 2 1 2 1 2
1 2 2  2 3  2 2





 3  2 2
1 2
1
1 2 1 2 1 2
1 2 1 2
20  5  18
20  5  18 1  48



1  48
1  48
1  48


20  5  18 1  48
1 

48 1  48
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

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
20  20. 48  5  5. 48  18  18. 48
1  48

4. 5  4. 5. 16. 3  5  5. 16. 3  9. 2  9. 2. 16. 3
47

2 5  8 5. 3  5  4 5. 3  3 2  12 2. 3
47

5  4 5. 3  3 2  12 2. 3
47

5  4 15  3 2  12 6
47
1.4.4 Complex Numbers
As stated in Definition 1.4.5, a complex number is a number of the form z  x  iy, where
x, y 
and i  1
Definition 1.4.6
If z  x  iy is a complex numbers, then the conjugate of z , denoted by z , is the complex
numbers z  x  iy. .
Complex numbers can be added, subtracted, multiplied and divided as the next example
shows.
Example 1.4.7
Given that z1  2  i and z2  4  2i, evaluate each of the following expressing your answer
in the form x  iy :
(a) z1  z2
(b) z1  z2
(c) z1 z2
(d)
z1
z2
Solutions
(a) z1  z2  2  i  4  2i  (2  4)  i  2i  6  i
(b) z1  z2  2  i  (4  2i)  2  4  i  2i  2  3i
(c) z1 z2  (2  i)(4  2i)  8  4i  4i  2i 2  8  2  10  10  0i
(d)
z1
2i

z2 4  2i
To express it in the form x  iy, we multiply by a special “1” using the conjugate of the
complex number in the denominator.
z1
2  i 4  2i (2  i )(4  2i ) 6  8i 3 2




  i.
z2 4  2i 4  2i
42  4i 2
20
10 5
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1.5 BINARY OPERATIONS
An operation can be applied to two numbers to generate a third number. Such operations are
known as binary operations.
Definition 1.5.1
Let S  {a, b, c,...} be any set. The operation * is a binary operation on S if and only if to
every ordered pair (a, b), where a, b  S , there is assigned unique element a * b  S. We
indicate this assignment by the notation (a, b)  a * b.
Example 1.5.1
The usual operation of arithmetic ,  and  are binary operations on
operation on any two numbers from generates a number in .
because each
Example 1.5.2
1. Determine whether the operation * defined by
(b) a * b  a  b
(a) a * b  2a b
is a binary operation on
.
2. For a binary operation * on
defined by a * b  a  b  ab, find
(b) (1* 2) * 5
(a) 2*5
Solutions
1. (a) For any a, b 
a * b  2a b  since a  b 
 * is a binary operation.
(b) For any a, b  , a  b 
but for some numbers
a  b  . For example,
if a  1, b  2, a * b  1*2  1  2  1  i  .
 * is not a binary operation.
2. (a) a *b  a  b  ab
2*5  2  5  2(5)  7  10  3  .
(b) (1* 2) * 5  1  2  1(2)  * 5
 1* 5
 1  5 1
 5
 1
Definition 1.5.2
The binary operation * on a set S is commutative if and only if for every ordered pair (a, b)
of elements S
a *b  b * a.
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Definition 1.5.3
The binary operation * on a set S is associative if and only if for every a, b, c  S ,
a *  b * c    a * b  * c.
Example 1.5.3
1. For the following binary operations *, determine whether they are
(i) commutative or (ii) associative
(a) a * b  2a b on
(b) a * b  a 2  b2 on
.
defined by a *b  a  b , find
2. For the binary operation * on
(i) 1*( 1* 2) (ii) (1* 1) * 2
Solution
1. (a) (i) a * b  2a b  2b  a  b * a
 The binary operation is commutative.
b c
(ii) a *(b * c)  a *(2bc )  2a 2
ab
(a * b)* c  (2a b )* c  2(2 )c
 a *(b * c)  (a * b) * c
Thus, the binary operation is not associative.
a
*
b
 a 2  b2  b2  a 2  b * a
(b) (i)
 The binary operation is not commutative.
(ii) a *(b * c)  a *(b 2  c 2 )  a 2   b 2  c 2 
2
( a * b) * c  ( a 2  b 2 ) * c   a 2  b 2   c 2
2
 a *(b * c)  (a * b) * c
Thus, the binary operation is not associative.
2. (i) 1*(1* 2)  1*(1  2)  1*1  1  1  2
(ii) (1* 1)*2  1  (1)  *2  0*2  0  2  2
It can be shown that this binary operation is associative.
Definition 1.5.4
A binary operation * on a set S has an identity, denoted by e, if
a * e  a  e * a.
Examples of binary operations with an identity elements are  and  on
for any a, b 
a 0  a  0 a
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. This is because
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implying that 0 is an identity element of  on
and
a 1  a  1 a
implying that 1 is an identity element of  on
.
Definition 1.5.5
If * is a binary operation on a set S with identity e, then for a  S , the element a 1 , called
the inverse of a, exists if and only if
a * a 1  e  a 1 * a.
The inverse of a
under addition is  a since
a  (a)  0  a  a.
The inverse of a
under multiplication is
a
1
since
a
1
1
 1   a.
a
a
NOTE: The notation a 1 for the inverse of a  S does not necessarily mean
THE END!
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1
.
a