2. Set
Intuitively, a set is a well-defined collection of objects of
any kind, which we call the elements of a set.
A collection of ‘lovely flowers’ is not a set, because the
objects (flowers) to be included are not well-defined.
If we have a set, we say that some objects –
belong (or do not belong) to the set,
are (or are not) in the set.
We say also that sets consist of their elements.
A set is determined by its elements.
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3. Some examples of set
All IIM students admitted in 2024 batch
All IIM-C students admitted in 2024 batch
All IIM-C students admitted in 2024 batch with STEM
background
All IIM-C students admitted in 2024 batch with non-
STEM background
All students in this class whose name starts with “S”
All students in this class whose name starts with “Z”
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4. More examples of set
Vowels in English alphabet
Odd numbers between 1 – 20.
All even numbers between 1-20 that are divisible
by 3.
Solution of the equation: x2
– 5x + 6 = 0.
All Integers between 1 – 5 (inclusive).
All real numbers between 0 – 1 (inclusive).
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5. Set - membership
Notation: We denote sets with upper-case letters like
A,B,X; elements with lower-case letters like a, b, x.
The following notation is used to show set
membership.
means that x is a member of the set A.
means that x is not a member of the set A.
A
x
A
x
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6. Ways of Describing Sets
List the elements
verbal description:
“A is the set of all integers from 1 to 6, inclusive”
A is the set of all natural numbers that are less
than 7 (or, <= 6)
A= 1,2,3,4,5,6
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7. Some “standard” sets in mathematics
Z : set of integers
Z+ : set of positive integers
Z- : set of negative integers
Q : set of rational numbers
Q+ : the set of positive rational numbers
Q- : the set of negative rational numbers
R : the set of real numbers
R+ : the set of positive real numbers
R- : the set of negative real numbers
N : set of natural numbers
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8. What about these?
Top 5 cricket players in India
Best 3 novel writers in English
5 most hated subjects in your MBA curriculum!
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9. Specification of sets
There are 2 main ways to specify a set:
(1) by listing all members
(2) by stating a property of its elements
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10. Specification of sets – type 1
Roster notation (set enumeration):
Set of all positive integers that are less than 7:
{1,2,3,4,5,6}
Set of all positive integers that are less than 7 AND even
numbers:{2,4,6}
This representation is easy to understand and construct.
No repetition of elements. E.g., set of all letters in “COMMITTEE” ={‘C’, ‘O’,
‘M’, ‘I’, ‘T’,‘E’}.
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11. Specification of sets – type 2
Describing a set using some kind of
“formula” or “expression” that would build
all the elements of our set.
Easy to check if an element belongs to a set
or not.
This is called set builder notation.
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12. Revisit previous examples…
Write following sets using set-builder form
Set of all positive integers that are less than 7
For example, in the set {a, e, i, o, u}, all the
elements possess a common property, namely,
each of them is a vowel in the English alphabet,
and no other letter possess this property.
Denoting this set by V, we write
V = {x : x is a vowel in English alphabet}
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13. Some Special Sets
The Null Set or Empty Set: set with no elements,
often symbolized by
A possible example: set of students in this class
whose First Name starts with X.
(By the way, this won’t be a valid example in China!)
The Universal Set: set of all elements under
consideration, and is often symbolized by
“U”
If A = {a, b, c} B = {d, e} C = {f, g, h, i} then, U =
{a, b, c, d, e, f, g, h, i} can be taken as universal set.
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14. Some Special Sets
Finite and infinite set: A set which consists of a
finite number of elements (counted or listed) is
called a finite set otherwise, the set is called an
infinite set.
Equal sets: Two sets A and B are equal, written as
A = B, if and only if they have exactly the same
elements.
A = {a, b, c} , B = {a, c, b}, C = {b, c}
so, A=B ≠ C
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15. Cardinality of sets
The cardinality of set is the number of
elements in a set, written as |A| or n(A) for
set A.
Example: |Ø|=0
A={u, v, w, x, y, z}; |A|=6
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16. Some Special Sets
Singleton set: If cardinality of a set is 1, then that
set is called singleton set.
Equivalent Sets: Two sets A and B are said to be
equivalent if their cardinal number is same, i.e.,
n(A) = n(B).
• Equal sets are always equivalent.
• Equivalent sets may not be equal.
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17. Subset and Super Set
Subset and super set: If every element of a set A is also an
element of a set B, then we say that A is a subset of B and write A ⊆ B.
B is super set of A and denoted as B A.
⊇
• Any set is a subset of itself; S ⊆ S
• Empty set is subset of all sets
• All sets are subset of universal set.
Proper subset: If A is a subset of B and A≠ B, then we say A is a
proper subset of B and B is proper superset of A.
• There is at least one element in B which is not present in A.
• It is written as A ⸦ B (proper subset) or B A
⊃
Some Examples:
{2, 4, 6} ⸦ {1, 2, 3, 4, 5, 6, 7}
{1, 2, 8} {1, 2, 3, 4, 5, 6, 7}
⊄
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18. Power Set
Power set: The collection of all the subsets of a set is
called power set.
• It is normally denoted as P(A) for set A.
• The total number of elements in a power set is , if the set
has n elements.
• For A = {a, b, c}, P(A)={Ø,{a},{b},{c},{a,b},{a,c},
{b,c},{a,b,c}}
• P(Ø)={Ø}
• If a set contains ‘n’ elements, then the number of subsets of
the set is
• If a set contains ‘n’ elements, then the number of proper
subsets of the set is – 1
• |P(A)|= 18
19. Venn Diagram
Most of the relationships between sets can be represented by
means of diagrams which are known as Venn diagrams. Venn
diagrams are named after the English logician, John Venn
(1834-1883). These diagrams consist of rectangles and closed
curves usually circles. The universal set is represented usually
by a rectangle and its subsets by circles. In Venn diagrams,
the elements of the sets are written in their respective circles.
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20. Operation on Sets
When two or more sets combine together to form one set
under the given conditions, then operations on sets are carried
out.
The basic operations are:
1. Union of Sets
2. Intersection of sets
3. Difference of sets
4. Complement of the Set
5. Cartesian Product of sets
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21. Union of Sets
• Union of two or more sets contains all the elements of all
the sets. Union symbol is usually a U
• Formal definition for the union of two sets:
A U B = {x | x ∈ A or x ∈ B}
Examples:
• {1, 2, 3} U {3, 4, 5} = {1, 2, 3, 4, 5}
• {New York, Washington} U {3, 4} = {New York,
Washington, 3, 4}
• {1, 2} U Ø = {1, 2}
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22. Union of Sets
Properties of the union operation:
• A U Ø = A (Identity law)
• A U U = U (Domination law)
• A U A = A (Idempotent law)
• A U B = B U A (Commutative law)
• A U (B U C) = (A U B) U C (Associative law)
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23. Intersection of Sets
• Intersection of two or more sets contains only the common
elements of all the sets; those items who are element of all the
sets for which we are finding out the intersection set.
• Intersection symbol is a ∩.
• Formal definition for the intersection of two sets:
A ∩ B = { x | x ∈ A and x ∈ B }
Examples
• {1, 2, 3} ∩ {3, 4, 5} = {3}
• {New York, Washington} ∩ {3, 4} = Ø
• {1, 2} ∩ Ø = Ø
• Any set intersection with the empty set yields the empty set
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24. Intersection of Sets
Properties of the intersection operation:
• A ∩ U = A (Identity law)
• A ∩ Ø = Ø (Domination law)
• A ∩ A = A (Idempotent law)
• A ∩ B = B ∩ A (Commutative law)
• A ∩ (B ∩ C) = (A ∩ B) ∩ C (Associative law)
• A ∩ (B C) = (A ∩ B) (A ∩ C) (Distributive law)
∪ ∪
• A (B ∩ C) = (A B) ∩ (A C)
∪ ∪ ∪
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25. Difference of Sets
A difference of two sets is the elements in one set that are NOT in
the other
• Difference symbol is a minus sign
• Formal definition for the difference of two sets:
A - B = { x | x ∈ A and x ∉ B }
Examples
• {1, 2, 3} - {3, 4, 5} = {1, 2}
• {New York, Washington} - {3, 4} = {New York, Washington}
• {1, 2} - Ø = {1, 2}
• The difference of any set S with the empty set will be the set S
• A-B ≠ B-A (Unless A=B)
• A-B= A- A∩B
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26. Complement of Set
A complement of a set is all the elements that are NOT in the set.
• Complement of set A is written as: Ā or A’ or
• Formal definition for the complement of a set: A = { x | x ∉
A } Or, U – A, where U is the universal set.
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27. Complement of Set
Some Properties of Complement of Set:
Complement laws: (i) A A′ = U (ii) A ∩ A′ =
∪ Ø
De Morgan’s law: (i) (A B)´ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ B′
∪ ∪
Law of double complementation : (A′)′ = A
Laws of empty set and universal set: Ø ′ = U and U′ = Ø
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28. Cartesian Product of Sets
Cartesian product of two sets is the set of all ordered pairs in which
first member is from first set and the second one is from second set.
• Denoted by A x B, and uses parenthesis
Example:
Given A = { a, b } and B = { 0, 1 }, what is their Cartesian product?
C = A x B = { (a,0), (a,1), (b,0), (b,1) }
• Formal definition of a Cartesian product:
A x B = { (a, b) | a ∈ A and b ∈ B }
• There can be Cartesian products on more than two sets
• A x B ≠ B x A
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29. Practical Problems on Union and
Intersection of Two Sets
Some Properties:
• n (A B) = n(A) + n(B) - n(A ∩ B)
∪
• If A ∩ B = Ø , then n(A B) = n(A) + n(B)
∪
• n (A B C) = n (A) + n (B) + n (C) – n (A ∩ B) – n (A ∩ C) –
∪ ∪
n (B ∩ C) + n (A ∩ B ∩ C)
• n(A - B) = n(A) - n(A ∩ B)
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30. Practical Problems on Union and
Intersection of Two Sets
Practice Problem:
In a survey of 400 students in a school, 100 were listed as taking
apple juice, 150 as taking orange juice and 75 were listed as taking
both apple as well as orange juice. Find how many students were
taking neither apple juice nor orange juice.
Ans: 225 students
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Editor's Notes
#10:In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.
#11:In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
