The document provides an overview of set theory, defining sets as well-defined collections of distinct objects known as elements. It explains various representations of sets, including tabular, descriptive, and set-builder forms, as well as key concepts like subsets, proper subsets, equal sets, power sets, and Cartesian products. Additionally, it covers operations on sets such as union and intersection, and addresses the distinction between finite and infinite sets.

1. SET THEORY
Lecture # 08

2. SETS
 A well defined collection of {distinct} objects is
called a set.
 The objects are called the elements or
members of the set.
 Sets are denoted by capital letters A, B, C …,
X, Y, Z.
 The elements of a set are represented by
lower case letters
a, b, c, … , x, y, z.

3.  If an object x is a member of a set A we write x
 A, which reads “x belongs to A” or “x is
in A” or “x is an element of A”,
 otherwise we write x  A, which reads “x does
not belong to A” or “x is not in A” or “x is not
an element of A”.

4. TABULAR FORM
 Listing all the elements of a set, separated by commas
and enclosed within braces or curly brackets{ }.
 EXAMPLES
In the following examples we write the sets
in Tabular Form.
A = {1, 2, 3, 4, 5} is the set of first five
Natural Numbers.
B = {2, 4, 6, 8, …, 50} is the set of Even
numbers up to 50.
C = {1, 3, 5, 7, 9, …} is the set of positive
odd numbers.
NOTE:
The symbol “…” is called an ellipsis. It is a short
for “and so fort”

5. DESCRIPTIVE FORM
 Stating in words the elements of a set.
 EXAMPLES
A = set of first five Natural Numbers.
B = set of positive even integers less or
equal to fifty.
C = set of positive odd integers.

6. SET BUILDER FORM
 Writing in symbolic form the common
characteristics shared by all the elements of the
set.
 EXAMPLES:
A = {x N / x<=5}
B = {x  E / 0 < x <=50}
C = {x  O / 0 < x }

7. SETS OF NUMBERS
Set of Natural Numbers
N = {1, 2, 3, … }
 Set of Whole Numbers
W = {0, 1, 2, 3, … }
 Set of Integers
Z = {…, -3, -2, -1, 0, +1, +2, +3, …}
= {0, 1, 2, 3, …}
 Set of Even Integers
E = {0,  2,  4,  6, …}
 Set of Odd Integers
O = { 1,  3,  5, …}
Set of Boolean Numbers
B = {true, false}

8.  Set of Prime Numbers
P = {2, 3, 5, 7, 11, 13, 17, 19, …}
 Set of Rational Numbers (or Quotient of
Integers)
Q = {x | x = p/q ; p, q Z, q  0}
 Set of Irrational Numbers
= Q/ = { x | x is not rational}
For example, 2, 3, , e, etc.
 Set of Real Numbers
R = Q  Q/
 10. Set of Complex Numbers
C = {z | z = x + iy; x, y  R}

10. SUBSET
 If A & B are two sets, A is called a subset of B,
written A  B, if, and only if, every element of A is
also an element of B.
Symbolically:
A  B  if x  A then x  B
 REMARK:
 When A  B, then B is called a superset of A.
 When A is not subset of B, then there exist at
least one x  A such that x B.
 Every set is a subset of itself.

11. EXAMPLES
 Let
A = {1, 3, 5} B = {1, 2, 3, 4, 5}
C = {1, 2, 3, 4} D = {3, 1, 5}
Then
A  B
C  B
A  D
D  A
A  C

12. EXAMPLES
 Let
A = {1, 3, 5} B = {1, 2, 3, 4, 5}
C = {1, 2, 3, 4} D = {3, 1, 5}
Then
A  B ( Because every element of A is in B )
C  B ( Because every element of C is also
an element of B )
A  D ( Because every element of A is also
an element of D and also note that every element of
D is in A so D  A )and A is not subset of C .
( Because there is an element 5 of A which is not in C
)

13. EXAMPLE
 The set of integers “Z” is a subset of the set of
Rational Number “Q”, since every integer ‘n’ could
be written as:
 Hence Z  Q.
Q
1
n
n 


14. PROPER SUBSET
 Let A and B be sets. A is a proper subset of B, if,
and only if, every element of A is in B but there is
at least one element of B that is not in A, and is
denoted as:
 Symbolically
A  B

15. EXAMPLE
 Let A = {1, 3, 5} B = {1, 2, 3, 5}
then A  B ( Because there is an element 2 of B
which is not in A)
N  W
W  Z
Z  Q
Q  R
N  W  Z  Q  R

16. EQUAL SETS
 Two sets A and B are equal if, and only if, every
element of A is in B and every element of B is in
A and is denoted A = B.
Symbolically:
A = B iff A  B and B  A
 A = {1, 2, 3, 4}
 B = {2, 4, 1, 3} A = B

17. EXAMPLE
 Let
A = {1, 2, 3, 6}
B = the set of positive divisors of 6
C = {3, 1, 6, 2}
D = {1, 2, 2, 3, 6, 6, 6}
Are A,B,C,D Equal sets?

18. EXAMPLE
 Let
A = {1, 2, 3, 6}
B = the set of positive divisors of 6
C = {3, 1, 6, 2}
D = {1, 2, 2, 3, 6, 6, 6}
 Then A, B, C, and D are all equal sets.

19. EXAMPLE
 Determine which of the following sets are equal
sets:
 (i) A = {55, 32, 77, 1} and B = {1, 32, 55, 77}
 (ii) X = {x : x is a prime number and 2<x<11} and Y
= {7, 3, 5}
 (iii) S = {2, 4, 6, 8} and T = {2, 4, 6}

20. SOLUTION
(i) To determine the set equality, we have to consider two things;
set elements and set cardinality. The cardinality of set A and B:
|A| = 4
And,
|B| = 4
So,
|A| = |B|
Both sets A and B have the same elements, which are 1, 32, 55,
and 7.
Hence, sets A and B are equal sets.

21. (II) TO DETERMINE SET EQUALITY, LET’S FIRST
SIMPLIFY SET X.
X = {x : x is a prime number and 2<x<11}
So,
X = {3, 5, 7}
Now, let’s find the cardinality.
|X| = 3
And,
|Y| = 3
So,
|X| = |Y|
Also, both sets have the same elements, which are 3,
5, and 7. Hence, sets X and Y are equal sets.

22. (III) TO DETERMINE SET EQUALITY, LET’S
CALCULATE THE CARDINALITY FIRST.
|S| = 4
And,
|T| = 3
As
|S| ≠ |T|
So the two sets, S and T, are not equal
sets.

23. POWER SET
 Given a set S, the power set of S is the set of all
subsets of the set.
 The power set of S is denoted by P(S).
If |A|=n, then |P(A)|=2n.

24. POWER SET
 If A = {1, 2}, then P(A) = { { }, {1}, {2}, {1, 2} }
Observe that A has 2 elements and P(A) has 22 = 4
elements.
 If A = {a, b, c}, then P(A) = { { }, {a}, {b}, {c}, {a, b},
{b, c}, {c, a}, {a, b, c} }
Observe that A has 3 elements and P(A) has 23 = 8
elements.

25. EXAMPLE
 What is the power set of the set {0, 1, 2}?
Solution: ?

26. EXAMPLE
 What is the power set of the set {0, 1, 2}?
Solution:
P({0, 1, 2}) = { , {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2},
{0, 1, 2} }

27. WHAT IS THE POWER SET OF AN EMPTY SET?
Solution: The power set of the empty set is an
empty set only.
Therefore, the power set of the empty set is an
empty set with one element, i.e., 20 = 1.
So, P(E) = {}.

28. CARTESIAN PRODUCT
 Let A and B be sets. The Cartesian Product of A
and B, denoted by A x B, is the set of all order pairs
(a, b), where aA and bB. Hence,
A x B = { (a, b) | aA  bB }
Note:
 The Cartesian Product A x B and B x A are not equals,
unless A =  or B = . So A x B = 

29. EXAMPLE
 What is the Cartesian Product of A = {1, 2} and
B = {a, b, c}?
Solution:
The Cartesian Product of A x B is
A x B = { (1, a), (1, b), (1, c). (2, a), (2, b), (2, c) }.

30. EXAMPLE
 Find the Cartesian product of A × B
A = {1, 2} and B = {4, 5, 6}
 Find the Cartesian product of A × B × C
A = {2,3} , B = {x,y}, and C = {5,6}

31. EXAMPLE
 Let A = {1, 2} and B = {4, 5, 6}
 A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6)}
 A = {2,3} , B = {x,y}, and C = {5,6}
 Cartesian product of A × B × C
 let us find the Cartesian product of A × B first.
 A × B = {(2,x), (2,y),(3,x),(3,y)}.
 A × B × C = {(2,x,5), (2,x,6), (2,y,5), (2,y,6), (3,x,5),
(3,x,6), (3,y,5), (3,y,6)}

32. NULL SET
 A set which contains no element is called a null
set, or an empty set or a void set.
 It is denoted by the Greek letter  (phi) or { }.
 EXAMPLE
A = {x|x is a person taller than 10 feet}
A =  ( Because there does not exist any human
being which is taller then 10 feet )
B = {x|x² = 4, x is odd}
B =  (Because we know that there does not exist
any odd whose square is 4)

33. UNIVERSAL SET
 The set of all elements under consideration is
called the Universal Set.
The Universal Set is usually denoted by U.

34. VENN DIAGRAM
 A Venn diagram is a graphical representation of
sets by regions in the plane.
 In Venn diagram the Universal Set U, which
contains all the objects under consideration, is
represented by rectangle.
A B
U

35. FINITE AND INFINITE SETS
 A set S is said to be finite if it contains exactly m
distinct elements where m denotes some non
negative integer.
 In such case we write
S = m or n(S) = m
CARDINALITY:
The cardinality of a finite set S, denoted by S, is
the number of (distinct) element of S.

36. EXAMPLES:
 The set S of letters of English alphabets is finite
and S = 26 (Cardinality)
 The null set  has no elements, is finite and
 = 0 (Cardinality)
 The set of positive integers {1, 2, 3,…} is
infinite.

37. EXERCISE:
 A = {month in the year}
 B = {even integers}
 C = {positive integers less than 1}
 D = {animals living on the earth}
 E = {lines parallel to x-axis}
 F = {x R  x100 + 29x50 – 1 = 0}
 G = {circles through origin}

38. EXERCISE:
 A = {month in the year}
 B = {even integers}
 C = {positive integers less than 1}
 D = {animals living on the earth}
 E = {lines parallel to x-axis}
 F = {x R  x100 + 29x50 – 1 = 0}
 G = {circles through origin}
FINITE
INFINITE
FINITE
FINITE
INFINITE
INFINITE
FINITE

39. MEMBERSHIP TABLE
 A table displaying the membership of elements
in sets. To indicate that an element is in a set, a
1 is used; to indicate that an element is not in a
set, a 0 is used.
A Ac
1 0
0 1

40. SET OPERATIONS

41. UNION
 Let A and B be subsets of a universal set U. The
union of sets A and B is the set of all elements
in U that belong to A or to B or to both, and is
denoted A  B.
 Symbolically:
A  B = {x U | x A or x  B}

42. EMAMPLE
 Let
U = {a, b, c, d, e, f, g}
A = {a, c, e, g},
B = {d, e, f, g}
Then A  B = {x U | x A or x  B}
= {a, c, d, e, f, g}

43. VENN DIAGRAM FOR UNION
 REMARK
1. A  B = B  A (that is union is commutative)
2. A  A  B and B  A  B
A
B
U
A  B is shaded

44. MEMBERSHIP TABLE FOR UNION
A B A  B
1 1 1
1 0 1
0 1 1
0 0 0

45. INTERSECTION
 Let A and B subsets of a universal set U. The
intersection of sets A and B is the set of all
elements in U that belong to both A and B and is
denoted A  B.
 Symbolically:
A  B = {x U | x  A and x B}

46. EXMAPLE
 Let U = {a, b, c, d, e, f, g}
A = {a, c, e, g},
B = {d, e, f, g}
Then A  B = {e, g}
U
A
B
A  B is shaded

47. REMARK
 A  B = B  A
 A  B  A and A  B  B
 If A  B = , then A & B are called disjoint sets.
 MEMBERSHIP TABLE FOR INTERSECTION:
 REMARK:
This membership table is similar to the truth table for
logical connective, conjunction ().
A B A  B
1 1 1
1 0 0
0 1 0
0 0 0

48. DIFFERENCE
 Let A and B be subsets of a universal set U. The
difference of “A and B” (or relative complement
of B in A) is the set of all elements in U that belong
to A but not to B, and is denoted
A – B or AB.
 Symbolically:
A – B = {x U | x  A and x B}

49. EXAMPLE:
 Let U = {a, b, c, d, e, f, g}
A = {a, c, e, g},
B = {d, e, f, g}
Then A – B = {a, c}
 VENN DIAGRAM FOR SET DIFFERENCE:
U
A-B is shaded
A B

50. REMARK:
 A – B  B – A set difference is not commutative.
 A – B  A
 A – B, A  B and B – A are mutually disjoint sets.
 MEMBERSHIP TABLE FOR SET DIFFERENCE:
 REMARK:
The membership table is similar to the truth table for ~ (p
q).
A B A – B
1 1 0
1 0 1
0 1 0
0 0 0

51. COMPLEMENT
 Let A be a subset of universal set U. The
complement of A is the set of all element in U
that do not belong to A, and is denoted A‫׳‬, A or Ac
 Symbolically:
Ac = {x U | x A}

52. EXAMPLE
 Let U = {a, b, c, d, e, f, g}
A = {a, c, e, g}
Then Ac = {b, d, f}
 VENN DIAGRAM FOR COMPLEMENT:
U
A
Ac is shaded
Ac

53. REMARK
 Ac = U – A
 A  Ac = 
 A  Ac = U
 MEMBERSHIP TABLE FOR COMPLEMENT:
 REMARK
This membership table is similar to the truth table for
logical connective negation (~).
A Ac
1 0
0 1

54. EXERCISE
 Given the following universal set U and its two
subsets P and Q, where
U = {x | x  Z, 0  x  10}
P = {x | x is a prime number}
Q = {x | x² < 70}
1. Draw a Venn diagram for the above
2. List the elements in P‫׳‬ Q

55. SOLUTION
 U = {x | x Z, 0  x  10}
U= {0, 1, 2, 3, …, 10}
 P = {x | x is a prime number}
P = {2, 3, 5, 7}
 Q = {x | x2 < 70}
Q= {0, 1, 2, 3, 4, 5, 6, 7, 8}

56. VENN DIAGRAM
 Pc  Q = ?
Pc = U – P = {0, 1, 2, 3, …, 10}- {2, 3, 5, 7}
= {0, 1, 4, 6, 8, 9, 10}
and
Pc  Q = {0, 1, 4, 6, 8, 9, 10}  {0, 1, 2, 3, 4, 5, 6, 7, 8}
= {0, 1, 4, 6, 8}
U
Q
P
2,3,5,7
0,1,4,6,8
9,10

57. EXERCISE
 Let
U = {1, 2, 3, 4, 5}, C = {1, 3}
Where A and B are non empty sets. Find A in
each of the following:
1) A  B = U, A  B =  and B = {1}
2) A  B and A  B = {4, 5}
3) A  B = {3}, A  B = {2, 3, 4} and B
 C = {1,2,3}
4) A and B are disjoint, B and C are disjoint, and
the union of A and B is the set {1, 2}.

58. SOLUTION
1. A  B = U, A  B =  and B = {1}
Since A  B = U
= {1, 2, 3, 4, 5}
and A  B = ,
Therefore
A = Bc
= {1}c
= {2, 3, 4, 5}

59. SOLUTION
2) A  B and A  B = {4, 5}
When A  B,
then A  B =
= {4, 5}
Also A being a proper subset of B implies
A = {4} or A = {5}

60. SOLUTION
3. A  B = {3}, A  B = {2, 3, 4} and B
 C = {1,2,3}
So
A = {3, 4} and B = {2, 3}
A
B
C
2
4
1

61. SOLUTION
4) A and B are disjoint, B and C are disjoint, and the
union of A and B is the set {1, 2}.
A  B = , B  C = , A  B = {1, 2}.
Also C = {1, 3}
A = {1}
A B
C
A B
C
2
A B
C
2
3
A B
C
2
3 4,5

62. EXERCISE:
 Use a Venn diagram to represent the following:
 (A  B)  Cc
 Ac  (B  C)
 (A – B)  C
 (A  Bc)  Cc
U
A
B
C
1
2
3
4
5
6
7 8
1

63. 1). (A  B)  CC
1
2
3
4
5 6
7
8
A
B
C
(A  B)  Cc is shaded
U

64. 2). AC  (B  C)
 Ac  (B  C) is shaded.
A
B
C
1
3
7
2
4
5 6
U
8

65. 3). (A – B)  C
U
A
B
C
1
2
3
4
5 6
7
8
(A – B)  C is shaded

66. 4). (A  BC)  CC
 (A  Bc)  Cc is shaded.
U
8
1
2
3
4
5 6
7
A
B
C

67. PROVING SET IDENTITIES BY VENN
DIAGRAMS
 Prove the following using Venn Diagrams:
 A – (A – B) = A  B
 (A  B)c = Ac  B c
 A – B = A  B c

68. SOLUTION
A - (A – B) = A  B
U
A
B
1 2 3
4
(a)
A – B is shaded
A = { 1, 2 }
B = { 2, 3 }
A – B ={ 1 }

69. (b)
A – (A – B) is shaded
1 2
3
4
U
A B A = { 1, 2 }
A – B = { 1 }
A – (A – B) = { 2 }

70. SOLUTION (II)
(A  B)c = Ac  Bc
U
A B
1
2
3
A  B
(a)
4

71. (b)
4
(A  B)c
U
1 2 3
B
A

72. A
4
U
Ac is shaded.
(c)
A
4
U
Bc is shaded.
(d)
3
2
1
1 2 3
B
A
(e)
4
Ac  Bc is shaded.
U
1
2 3
A B
B
A

73.  Now diagrams (b) and (e) are same hence
RESULT: (A  B)c = Ac  Bc

74. SOLUTION (III)
A – B = A  Bc

75. U
A B
1 2 3
4
(a)
A – B is shaded.
A
4
U
A  Bc is shaded
(c)
3
2
1
A
1 2 3
B U
4
Bc is shaded.
(b)
B
A

76.  From diagrams (a) and (b) we can say
RESULT: A – B = A  Bc