Lecture-10
Set Theory
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What is set ?
The collection of well-defined distinct objects is known as a set.
The word well-defined refers to a specific property which makes it
easy to identify whether the given object belongs to the set or not.
The word ‘distinct’ means that the objects of a set must be all
different.
Examples of a set:
•The collection of children in class VII whose weight exceeds 35 kg
represents a set.
•The set of all even numbers
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Notation of a Set:
A set is usually denoted by capital letters and elements
are denoted by small letters
If x is an element of set A, then we say x ϵ A. [x belongs
to A]
If x is not an element of set A, then we say x ∉ A. [x
does not belong to A]
For example:
The collection of vowels in the English alphabet.
Let us denote the set by V, then the elements of the set
are a, e, i, o, u or we can say, V = [a, e, i, o, u].
We say a ∈ V, e ∈ V, i ∈ V, o ∈ V and u ∈ V.
Also, we can say b ∉ V, c ∉ v, d ∉ v, etc.
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Different notations in sets
∈
∉
: or |
∅ or { }
n(A)
∪
∩
⊆
⊄
∀
∃
{,}
Belongs to
Does not belongs to
Such that
Null set or empty set
Cardinal number of the set A
Union of two sets
Intersection of two sets
Subset of a set
Not a subset
Universal set
For all
There exists
Set braces
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Empty Set or Null Set:
A set which does not contain any element is called an empty set, or
the null set or the void set and it is denoted by ∅ and is read as phi. In
roster form, ∅ is denoted by {}. An empty set is a finite set, since the number of
elements in an empty set is finite, i.e., 0.
For example:
1) The set of natural numbers less than 0. Clearly there is no whole
number less than 0. Therefore, it is an empty set.
2)N = {x : x ∈ N, 3 < x < 4} and A = {x : 2 < x < 3, x is a natural
number}
Here A is an empty set because there is no natural number between
2 and 3 as well as 3 and 4
3)Let B = {x : x is a composite number less than 4}.
Here B is an empty set because there is no composite number less
than 4.
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Singleton Set:
A set which contains only one element is called a singleton set.
For example:
• A = {x : x is neither prime nor composite}
It is a singleton set containing one element, i.e., 1.
• B = {x : x is a whole number, x < 1}
This set contains only one element 0 and is a singleton set.
• Let A = {x : x ∈ N and x² = 4}
Here A is a singleton set because there is only one element 2 whose square is
4.
• Let B = {x : x is a even prime number}
Here B is a singleton set because there is only one prime number
which is even, i.e., 2.
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Finite Set:
A set which contains a definite number of elements is called a finite
set. Empty set is also called a finite set.
For example:
• The set of all colors in the rainbow.
• N = {x : x ∈ N, x < 7}
• P = {2, 3, 5, 7, 11, 13, 17, ...... 97}
Infinite Set:
The set whose elements cannot be listed, i.e., set containing neverending elements is called an infinite set.
For example:
• Set of all points in a plane
• A = {x : x ∈ N, x > 1}
• Set of all prime numbers
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Subset of a set
Set A is said to be a subset of Set B if all the elements of Set A are also
present in Set B. In other words, set A is contained inside Set B.
Example: If set A has {X, Y} and set B has {X, Y, Z}, then A is the subset of
B because elements of A are also present in set B.
Subset Symbol
In set theory, a subset is denoted by the symbol ⊆ and read as ‘is a
subset of’.
Using this symbol we can express subsets as follows:
A ⊆ B; which means Set A is a subset of Set B.
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Proper subset
• A set A is a proper subset of another set B if A
is a subset and B and A is not equal to B.
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Representing Sets
1) Venn Diagrams
2) Roster Form
3) Set Builder Notation
1) Venn diagram is a pictorial representation of sets by sets of points in the plane.
The universal set is usually represented by the interior of a rectangle
U
B
A
A
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2) Roster Form
In this form, all the elements are enclosed within braces {} and they are
separated by commas (,).
For example:
A collection of all the numbers found on a dice N = {1, 2, 3, 4, 5, 6}.
Properties of roster form:
The order in which the elements are listed in the Roster form for any
Set is immaterial.
The dots at the end of the last element of any Set represent its infinite
form and indefinite nature.
For example:
V = {a, e, i, o, u} is same as V = {u, o, e, a, i}
Group of odd natural numbers = {1, 3, 5, …}
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3) Set Builder Form
In this form, all the elements possess a single common property which is NOT featured
by any other element outside the Set.
For example: a group of vowels in English alphabetical series. The representation is
done as follows.
Let V be the collection of all English vowels,
then V = {x: x is a vowel in English alphabetical series.}
Properties of Roster form:
•Colon (:) is a mandatory symbol for this type of representation.
•After the colon sign, we write the common characteristic property possessed by ALL
the elements belonging to that Set and enclose it within braces.
•If the Set doesn’t follow a pattern, its Set builder form cannot be written.
More examples:
D = {x: x is an integer and – 3 < x < 19}
O = {y: y is a natural number greater than 5}
I = {f: f is a two – digit prime number less than 100}
R = {s: s is a natural number such that sum of its digits is 4}
X = {m: m is a positive integer < 40}
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What are the five basic operations on sets?
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Difference of Sets
5. Cartesian Product of sets
1) Union of Sets
The union of two sets A and B, denoted as A∪B, results in a set that includes all
elements belonging to either A or B.
A∪B={x:x∈A or x∈B}
Example
Consider the set A={1,2,3} and the set B={2,3,4}.
Then, A∪B={1,2,3,4}.
Let set A = {a,b,c,d} and set B = {e,f}
Then, A∪B = {a,b,c,d,e,f}
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2) Intersection of Sets
The intersection of two sets A and B, denoted as A∩B, results in a set
that includes all elements belonging to both the sets.
A∩B={x:x∈A and x∈B}
Example
Consider the set A={1,2,3} and set B={2,3,4}. Then, A∩B={2,3}
 Let A = {2, 4, 6, 8, 10} and B = {1, 3, 8, 4, 6}. Find intersection of two set A and B.
Solution:
A ∩ B = {4, 6, 8}
Therefore, 4, 6 and 8 are the common elements in both the sets.
If X = {a, b, c} and Y = {ф}. Find intersection of two given sets X and Y.
Solution:
X∩Y={}
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3) Complement of a Set Definition
If U is a universal set and A be any subset of U then the complement
of A is the set of all members of the universal set U which are not the
elements of A.
A′ = {x : x ∈ U and x ∉ A}
Alternatively it can be said that the difference of the universal
set U and the subset A gives us the complement of set A.
Example :
If A = { 1, 2, 3, 4} and U = { 1, 2, 3, 4, 5, 6, 7, 8} then
find A complement ( A’).
Solution :
A = { 1, 2, 3, 4} and Universal set = U = { 1, 2, 3, 4, 5,
6, 7, 8}
Complement of set A contains the elements present in universal set but not in set A.
Elements are 5, 6, 7, 8.
∴ A complement = A’ = { 5, 6, 7, 8}.
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4) Difference of sets
Given set A and set B the set difference of set B from set A is the set of
all element in A, but not in B.
We can write as A − B
Example :
1) Let A = {1, 2, 4, 6, 7, 8, 9 } and B = { 1, 2, 4, 6} Find A − B and B – A
Solution: What we see in A that are not in B are 7, 8, and 9 Hence A − B = { 7, 8, 9}
2)Let A = {3 , 4 , 8 , 9 , 11 , 12 } and B = {1 , 2 , 3 , 4 , 5 }. Find A – B and B – A.
Solution: We can say that
A – B = {8 , 9 , 11 , 12} as these elements belong to A but not to B
B – A ={1,2,5} as these elements belong to B but not to A.
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5) Cartesian Product of Two Sets
Suppose that A and B are non-empty sets. The Cartesian product of two
sets A and B, denoted A×B, is the set of all possible ordered
pairs (a,b), where a∈A and b∈B:
Hence A×B={(a,b)∣a∈A and b∈B}.
The Cartesian product is also known as the cross product.
Example:
If A = { 1, 3, 5} and B = {2, 3}, then Find: (i) A × B (ii) B × A (iii) A × A (iv) (B × B)
Solution:
A ×B={1, 3, 5} × {2,3} = [{1, 2},{1, 3},{3, 2},{3, 3},{5, 2},{5, 3}]
B × A = {2, 3} × {1, 3, 5} = [{2, 1},{2, 3},{2, 5},{3, 1},{3, 3},{3, 5}]
A × A = {1, 3, 5} × {1, 3, 5}= [{1, 1},{1, 3},{1, 5},{3, 1},{3, 3},{3, 5},{5, 1},{5, 3},{5, 5}]
B × B = {2, 3} × {2, 3} = [{2, 2},{2, 3},{3, 2},{3, 3}]
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Power Set
• Given a set S, the power set of S, written P(S) is the set
containing all subsets of set S.
• Example
let S= {1, 2, 3}
Then the power set is of the form;
P(S) = { { }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, S}
Cardinal number of a set A, n(A)
The cardinality of a set A is the number of all distinct
elements of set A.
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Disjoint sets
• Two sets are said to be disjoint if their
intersection is empty.
Properties
i) n(A ∩B) = 0
ii) n(AUB)=n(A)+n(B)
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Partition of a set
• A partition of a set A is a collection of non
empty subsets A1, A2,…An such that;
• i) A is the union of all the subsets.
• ii) The subsets are piecewise disjoint, that is
the intersection of any two subsets is empty.
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Laws of sets
•
•
•
•
•
•
•
•
Idempotent
Associative
Commutative
Distributive
Identity
Involution
Complement
DeMorgan’s Laws
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Excercises:
1) If P = { 1,3,5,7,9} and Q = { 2,3,5,7} Find P ∪ Q, and P ∩ Q
2) If Set A = {a,b,c,d,e,f,g,h,i} and Set B = {a,e,i,o,u}. Find A ∩ B.
3) Let P = {1, 2, 3, 5, 7, 11}, Q = {first five even natural numbers}. Find P ∩ Q and
(P ∪ Q).
4) Given that A = {1,3,5,7,9}, B = {0,5,10,15}, and U = {0,1,3,5,7,9,10,11,15,20}. Find
A ∩ B and A ∩ B'.
5) Consider the sets: A = {3, 4, 5} B = {2, 4, 6} C = {3, 6, 9} Find A U B U C
6) Let A, B and C be three sets such that:
Set A = {2, 4, 6, 8, 10, 12}, set B = {3, 6, 9, 12, 15} and set C = {1, 4, 7, 10, 13, 16}.
Find:
(i) A ∪ B (ii) A ∩ B (iii) B ∩ A (iv) B ∪ A (v) B ∪ C
(vi) Is A ∪ B = B ∪ A?
(vii) Is B ∩ C = B ∪ C?
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8) Given set A = {2, 4, 6, 8, 10, 12}, set B = {3, 6, 9, 12, 15, 18} and set C = {0, 6, 12, 18}
Find:
(i) A – B (ii) B – C (iii) C – A (iv) A – C
9) Given: P = {a, c, d , m}, Q = {c, e, m, x} and R = {a, e, i, o}
Find:
(i) P – R (ii) Q – P (iii) R – Q
10) Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 2, 4, 6, 8, 10} B = {1, 3, 5, 7, 8, 9}
Find:
(a) A'
(b) B'
(c) A' ∪ B'
(d) A' ∩ B'
(e) (A ∪ B)'
Also show (A ∪ B)' = A' ∩ B'.
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Write each of the following sets in the Roster form:
(i) The set of first seven natural numbers.
(ii) The set of whole numbers less than 5.
(iii) The set of five numbers each of which is divisible by 3.
(iv) The set of whole numbers less than 20 and divisible by 3.
(v) The set of integers greater than -2 and less than 4.
(vi) The set of integers between -4 and 4.
(vii) The set of letters in the word ‘mathematics’.
(viii) The set of consonants in the word ‘possession’.
(ix) The set of first three letters in the word ‘booklet’.
(x) {x : x is a letter in the word ‘SCHOOL’}.
(xi) {x : x is an odd natural number between 10 and 20}.
(xii) {Vowels used in the word ‘AMERICA’}
(xiii) {Consonants used in the word ‘MADRAS’}.
(ix) Set of prime factors of 36.
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Write each given set in the Set-Builder Form:
(i) {2, 4, 6, 8, 10}
(ii) {2, 3, 5, 7, 11}
(iii) {January, June, July}
(iv) {a, e, i, o, u}
(v) {Tuesday, Thursday}
(vi) {1, 4, 9, 16, 25}
(vii) {5, 10, 15, 20, 25, 30}
(viii) D = {violet, indigo, blue, green, yellow, orange, red}
(ix) E = {January, March, May, July, August, October, December}
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Answers:
(i) {x : x is an even natural numbers less than 12}
(ii) {x : x is a prime numbers less than 12}
(iii) {x : x is a month whose name starts with letter J}
(iv) { x : x is a vowel in English alphabets}
(v) {x : x is a day of the week whose name starts with letter T}
(vi) {x : x is a perfect square natural number upto 25}
(vii) {x : x is a natural number upto 30 and divisible by 5}
(viii)D = {x | x is a color in rainbow}.
(ix) E = {x | x is a month having 31 days}.
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Answers:
(i) {1, 2, 3, 4, 5, 6, 7}
(ii) {0, 1, 2, 3, 4}
(iii) {3, 6, 9, 12, 15}
(iv) {3, 6, 9, 12, 15, 18}
(v) {-1, 0, 1, 2, 3}
(vi) {-3, -2, -1, 0, 1, 2, 3}
(vii) {m, a, t, h, e, i, c, s}
(viii) {p, s, n}
(ix) {b, o, k}
(x) {s, c, h, o, l}
(xi) {11, 13, 15, 17, 19}.
(xii) {0, -1, -2, -3, ….}
(xiii) {0, 1, 4, 9, 16, ….}
(xiv) {a, e, i}
(xv) {m, d, r, s}
(xvi) {2, 3}
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