SETS in Discrete Structure

Discrete Structure & Theory of Logic Unit 1
G L Bajaj Group of Intitutions, Mathura | Compiled By: Nandini Sharma Page 1
What is set (in mathematics)?
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.
For example:
1. The collection of children in class VII whose weight exceeds 35 kg represents a set.
2. The collection of all the intelligent children in class VII does not represent a set because the word
intelligent is vague. What may appear intelligent to one person may not appear the same to another
person.
Elements of Set:
The different objects that form a set are called the elements of a set. The elements of the set are
written in any order and are not repeated. Elements are denoted by small letters.
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.
Solution :
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.
In representation of a set the following three methods are commonly used:
(i) Statement form method
(ii) Roster or tabular form method

(iii) Rule or set builder form method
1. Statement form:
In this, well-defined description of the elements of the set is given and the same are enclosed in curly
brackets.
For example:
(i) The set of odd numbers less than 7 is written as: {odd numbers less than 7}.
(ii) A set of football players with ages between 22 years to 30 years.
(iii) A set of numbers greater than 30 and smaller than 55.
(iv) A set of students in class VII whose weights are more than your weight.
2. Roster form or tabular form:
In this, elements of the set are listed within the pair of brackets { } and are separated by commas.
For example:
(i) Let N denote the set of first five natural numbers.
Therefore, N = {1, 2, 3, 4, 5} → Roster Form
(ii) The set of all vowels of the English alphabet.
Therefore, V = {a, e, i, o, u} → Roster Form
(iii) The set of all odd numbers less than 9.
Therefore, X = {1, 3, 5, 7} → Roster Form
(iv) The set of all natural number which divide 12.
Therefore, Y = {1, 2, 3, 4, 6, 12} → Roster Form
(v) The set of all letters in the word MATHEMATICS.
Therefore, Z = {M, A, T, H, E, I, C, S} → Roster Form
(vi) W is the set of last four months of the year.
Therefore, W = {September, October, November, December} → Roster Form
Note:

The order in which elements are listed is immaterial but elements must not be repeated.
3. Set builder form:
In this, a rule, or the formula or the statement is written within the pair of brackets so that the set is well
defined. In the set builder form, all the elements of the set, must possess a single property to become the
member of that set.
In this form of representation of a set, the element of the set is described by using a symbol ‘x’ or any other
variable followed by a colon The symbol ‘:‘ or ‘|‘ is used to denote such that and then we write the property
possessed by the elements of the set and enclose the whole description in braces. In this, the colon stands for
‘such that’ and braces stand for ‘set of all’.
For example:
(i) Let P is a set of counting numbers greater than 12;
the set P in set-builder form is written as :
P = {x : x is a counting number and greater than 12}
or
P = {x | x is a counting number and greater than 12}
This will be read as, 'P is the set of elements x such that x is a counting number and is greater than 12'.
Note:
The symbol ':' or '|' placed between 2 x's stands for such that.
(ii) Let A denote the set of even numbers between 6 and 14. It can be written in the set builder form
as;
A = {x|x is an even number, 6 < x < 14}
or A = {x : x ∈ P, 6 < x < 14 and P is an even number}
(iii) If X = {4, 5, 6, 7} . This is expressed in roster form.
Let us express in set builder form.
X = {x : x is a natural number and 3 < x < 8}
(iv) The set A of all odd natural numbers can be written as
A = {x : x is a natural number and x = 2n + 1 for n ∈ W}
Solved example using the three methods of representation of a set:

The set of integers lying between -2 and 3.
Statement form: {I is a set of integers lying between -2 and 3}
Roster form: I = {-1, 0, 1, 2}
Set builder form: I = {x : x ∈ I, -2 < x < 3}
What are the different notations in sets?
To learn about sets we shall use some accepted notations for the familiar sets of numbers.
Some of the different notations used in sets are:
∈
∉
: or |
∅
n(A)
∪
∩
N
W
I or Z
Z+
Q
Q+
R
R+
C
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
Set of natural numbers = {1, 2, 3, ……}
Set of whole numbers = {0, 1, 2, 3, ………}
Set of integers = {………, -2, -1, 0, 1, 2, ………}
Set of all positive integers
Set of all rational numbers
Set of all positive rational numbers
Set of all real numbers
Set of all positive real numbers
Set of all complex numbers
These are the different notations in sets generally required while solving various types of problems on sets.
Note:

(i) The pair of curly braces { } denotes a set. The elements of set are written inside a pair of curly braces
separated by commas.
(ii) The set is always represented by a capital letter such as; A, B, C, …….. .
(iii) If the elements of the sets are alphabets then these elements are written in small letters.
(iv) The elements of a set may be written in any order.
(v) The elements of a set must not be repeated.
(vi) The Greek letter Epsilon ‘∈’ is used for the words ‘belongs to’, ‘is an element of’, etc.
Therefore, x ∈ A will be read as ‘x belongs to set A’ or ‘x is an element of the set A'.
(vii) The symbol ‘∉’ stands for ‘does not belongs to’ also for ‘is not an element of’.
Therefore, x ∉ A will read as ‘x does not belongs to set A’ or ‘x is not an element of the set A'.
The standard sets of numbers can be expressed in all the three forms of representation of a set i.e.,
statement form, roster form, set builder form.
1. N = Natural numbers
= Set of all numbers starting from 1 → Statement form
= Set of all numbers 1, 2, 3, ………..
= {1, 2, 3, …….} → Roster form
= {x :x is a counting number starting from 1} → Set builder form
Therefore, the set of natural numbers is denoted by N i.e., N = {1, 2, 3, …….}
2. W = Whole numbers
= Set containing zero and all natural numbers → Statement form
= {0, 1, 2, 3, …….} → Roster form
= {x :x is a zero and all natural numbers} → Set builder form
Therefore, the set of whole numbers is denoted by W i.e., W = {0, 1, 2, .......}
3. Z or I = Integers
= Set containing negative of natural numbers, zero and the natural numbers
→ Statement form

= {………, -3, -2, -1, 0, 1, 2, 3, …….} → Roster form
= {x :x is a containing negative of natural numbers, zero and the natural numbers}
→ Set builder form
Therefore, the set of integers is denoted by I or Z i.e., I = {...., -2, -1, 0, 1, 2, ….}
4. E = Even natural numbers.
= Set of natural numbers, which are divisible by 2 → Statement form
= {2, 4, 6, 8, ……….} → Roster form
= {x :x is a natural number, which are divisible by 2} → Set builder form
Therefore, the set of even natural numbers is denoted by E i.e., E = {2, 4, 6, 8,.......}
5. O = Odd natural numbers.
= Set of natural numbers, which are not divisible by 2 → Statement form
= {1, 3, 5, 7, 9, ……….} → Roster form
= {x :x is a natural number, which are not divisible by 2} → Set builder form
Therefore, the set of odd natural numbers is denoted by O i.e., O = {1, 3, 5, 7, 9,.......}
Therefore, almost every standard sets of numbers can be expressed in all the three methods as discussed above.
What are the different types of sets?
The different types of sets are explained below with examples.
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: (a) The set of whole numbers less than 0.
(b) Clearly there is no whole number less than 0.
Therefore, it is an empty set.

(c) N = {x : x ∈ N, 3 < x < 4}
• Let 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.
• 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.
Note:
∅ ≠ {0} ∴ has no element.
{0} is a set which has one element 0.
The cardinal number of an empty set, i.e., n(∅) = 0
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.
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 never-ending 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
• B = {x : x ∈ W, x = 2n}
Note:
All infinite sets cannot be expressed in roster form.
For example:
The set of real numbers since the elements of this set do not follow any particular pattern.
Cardinal Number of a Set:
The number of distinct elements in a given set A is called the cardinal number of A. It is denoted by
n(A).
For example:
• A {x : x ∈ N, x < 5}
A = {1, 2, 3, 4}
Therefore, n(A) = 4

• B = set of letters in the word ALGEBRA
B = {A, L, G, E, B, R}
Therefore, n(B) = 6
Equivalent Sets:
Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The
symbol for denoting an equivalent set is ‘↔’.
For example:
A = {1, 2, 3} Here n(A) = 3
B = {p, q, r} Here n(B) = 3
Therefore, A ↔ B
Equal sets:
Two sets A and B are said to be equal if they contain the same elements. Every element of A is an
element of B and every element of B is an element of A.
For example:
A = {p, q, r, s}
B = {p, s, r, q}
Therefore, A = B
The various types of sets and their definitions are explained above with the help of examples.
The relations are stated between the pairs of sets. Learn to state, giving reasons whether the
following sets are equivalent or equal, disjoint or overlapping.
Equal Set:
Two sets A and B are said to be equal if all the elements of set A are in set B and vice versa. The
symbol to denote an equal set is =.
A = B means set A is equal to set B and set B is equal to set A.

For example;
A = {2, 3, 5}
B = {5, 2, 3}
Here, set A and set B are equal sets.
Equivalent Set:
Two sets A and B are said to be equivalent sets if they contain the same number of elements. The
symbol to denote equivalent set is ↔.
A ↔ means set A and set B contain the same number of elements.
For example;
A = {p, q, r}
B = {2, 3, 4}
Here, we observe that both the sets contain three elements.
Notes:
Equal sets are always equivalent.
Equivalent sets may not be equal.
Disjoint Sets:
Two sets A and B are said to be disjoint, if they do not have any element in common.
For example;
A = {x : x is a prime number}
B = {x : x is a composite number}.
Clearly, A and B do not have any element in common and are disjoint sets.
Overlapping sets:

Two sets A and B are said to be overlapping if they contain at least one element in common.
For example;
• A = {a, b, c, d}
B = {a, e, i, o, u}
• X = {x : x ∈ N, x < 4}
Y = {x : x ∈ I, -1 < x < 4}
Here, the two sets contain three elements in common, i.e., (1, 2, 3)
The above explanations will help us to find whether the pairs of sets are equal sets or equivalent sets,
disjoint sets or overlapping sets.
The relations are stated between the pairs of sets. Learn to state, giving reasons whether the
following sets are equivalent or equal, disjoint or overlapping.
Equal Set:
Two sets A and B are said to be equal if all the elements of set A are in set B and vice versa. The
symbol to denote an equal set is =.
A = B means set A is equal to set B and set B is equal to set A.
For example;
A = {2, 3, 5}
B = {5, 2, 3}
Here, set A and set B are equal sets.
Equivalent Set:
Two sets A and B are said to be equivalent sets if they contain the same number of elements. The
symbol to denote equivalent set is ↔.
A ↔ means set A and set B contain the same number of elements.
For example;
A = {p, q, r}

B = {2, 3, 4}
Here, we observe that both the sets contain three elements.
Notes:
Equal sets are always equivalent.
Equivalent sets may not be equal.
Disjoint Sets:
Two sets A and B are said to be disjoint, if they do not have any element in common.
For example;
A = {x : x is a prime number}
B = {x : x is a composite number}.
Clearly, A and B do not have any element in common and are disjoint sets.
Overlapping sets:
Two sets A and B are said to be overlapping if they contain at least one element in common.
For example;
• A = {a, b, c, d}
B = {a, e, i, o, u}
• X = {x : x ∈ N, x < 4}
Y = {x : x ∈ I, -1 < x < 4}
Here, the two sets contain three elements in common, i.e., (1, 2, 3)
The above explanations will help us to find whether the pairs of sets are equal sets or equivalent sets,
disjoint sets or overlapping sets.
Definition of Subset:
If A and B are two sets, and every element of set A is also an element of set B, then A is called a
subset of B and we write it as A ⊆ B or B ⊇ A

The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’
• Every set is a subset of itself, i.e., A ⊂ A, B ⊂ B.
• Empty set is a subset of every set.
• Symbol ‘⊆’ is used to denote ‘is a subset of’ or ‘is contained in’.
• A ⊆ B means A is a subset of B or A is contained in B.
• B ⊆ A means B contains A.
For example;
1. Let A = {2, 4, 6}
B = {6, 4, 8, 2}
Here A is a subset of B
Since, all the elements of set A are contained in set B.
But B is not the subset of A
Since, all the elements of set B are not contained in set A.
Notes:
If ACB and BCA, then A = B, i.e., they are equal sets.
Every set is a subset of itself.
Null set or ∅ is a subset of every set.
2. The set N of natural numbers is a subset of the set Z of integers and we write N ⊂ Z.
3. Let A = {2, 4, 6}
B = {x : x is an even natural number less than 8}
Here A ⊂ B and B ⊂ A.
Hence, we can say A = B
4. Let A = {1, 2, 3, 4}

B = {4, 5, 6, 7}
Here A ⊄ B and also B ⊄ C
[⊄ denotes ‘not a subset of’]
Super Set:
Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B ⊇ A.
Symbol ⊇ is used to denote ‘is a super set of’
For example;
A = {a, e, i, o, u}
B = {a, b, c, ............., z}
Here A ⊆ B i.e., A is a subset of B but B ⊇ A i.e., B is a super set of A
Proper Subset:
If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B. The
symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B.
For example;
1. A = {1, 2, 3, 4}
Here n(A) = 4
B = {1, 2, 3, 4, 5}
Here n(B) = 5
We observe that, all the elements of A are present in B but the element ‘5’ of B is not present in A.
So, we say that A is a proper subset of B.
Symbolically, we write it as A ⊂ B
Notes:
No set is a proper subset of itself.
Null set or ∅ is a proper subset of every set.
2. A = {p, q, r}

B = {p, q, r, s, t}
Here A is a proper subset of B as all the elements of set A are in set B and also A ≠ B.
Notes:
No set is a proper subset of itself.
Empty set is a proper subset of every set.
Power Set:
The collection of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A),
every element is a set.
For example;
If A = {p, q} then all the subsets of A will be
P(A) = {∅, {p}, {q}, {p, q}}
Number of elements of P(A) = n[P(A)] = 4 = 22
In general, n[P(A)] = 2m where m is the number of elements in set A.
Universal Set
A set which contains all the elements of other given sets is called a universal set. The symbol for
denoting a universal set is ∪ or ξ.
For example;
1. If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7}
then U = {1, 2, 3, 4, 5, 7}
[Here A ⊆ U, B ⊆ U, C ⊆ U and U ⊇ A, U ⊇ B, U ⊇ C]
2. If P is a set of all whole numbers and Q is a set of all negative numbers then the universal set is a
set of all integers.
3. 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.
Number of Subsets of a given Set:
If a set contains ‘n’ elements, then the number of subsets of the set is 2nn.
Number of Proper Subsets of the Set:
If a set contains ‘n’ elements, then the number of proper subsets of the set is 2nn - 1.
If A = {p, q} the proper subsets of A are [{ }, {p}, {q}]
⇒ Number of proper subsets of A are 3 = 222 - 1 = 4 - 1
In general, number of proper subsets of a given set = 2mm - 1, where m is the number of elements.
For example:
1. If A {1, 3, 5}, then write all the possible subsets of A. Find their numbers.
Solution:
The subset of A containing no elements - { }
The subset of A containing one element each - {1} {3} {5}
The subset of A containing two elements each - {1, 3} {1, 5} {3, 5}
The subset of A containing three elements - {1, 3, 5)
Therefore, all possible subsets of A are { }, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}
Therefore, number of all possible subsets of A is 8 which is equal 233.
Proper subsets are = { }, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}
Number of proper subsets are 7 = 8 - 1 = 233 - 1
2. If the number of elements in a set is 2, find the number of subsets and proper subsets.
Solution:
Number of elements in a set = 2
Then, number of subsets = 222 = 4
Also, the number of proper subsets = 222 - 1

= 4 – 1 = 3
3. If A = {1, 2, 3, 4, 5}
then the number of proper subsets = 255 - 1
= 32 - 1 = 31 {Take [2nn - 1]}
and power set of A = 255 = 32 {Take [2nn]}
Definition of operations on sets:
When two or more sets combine together to form one set under the given conditions, then operations
on sets are carried out.
What are the four basic operations on sets?
Solution:
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Definition of Union of Sets:
Union of two given sets is the smallest set which contains all the elements of both the sets.
To find the union of two given sets A and B is a set which consists of all the elements of A and all the
elements of B such that no element is repeated.
The symbol for denoting union of sets is ‘∪’.
For example;
Let set A = {2, 4, 5, 6}
and set B = {4, 6, 7, 8}
Taking every element of both the sets A and B, without repeating any element, we get a new set = {2, 4,
5, 6, 7, 8}
This new set contains all the elements of set A and all the elements of set B with no repetition of elements
and is named as union of set A and B.

The symbol used for the union of two sets is ‘∪’.
Therefore, symbolically, we write union of the two sets A and B is A ∪ B which means A union B.
Therefore, A ∪ B = {x : x ∈ A or x ∈ B}
Solved examples to find union of two given sets:
1. If A = {1, 3, 7, 5} and B = {3, 7, 8, 9}. Find union of two set A and B.
Solution:
A ∪ B = {1, 3, 5, 7, 8, 9}
No element is repeated in the union of two sets. The common elements 3, 7 are taken only once.
2. Let X = {a, e, i, o, u} and Y = {ф}. Find union of two given sets X and Y.
Solution:
X ∪ Y = {a, e, i, o, u}
Therefore, union of any set with an empty set is the set itself.
3. If set P = {2, 3, 4, 5, 6, 7}, set Q = {0, 3, 6, 9, 12} and set R = {2, 4, 6, 8}.
(i) Find the union of sets P and Q
(ii) Find the union of two set P and R
(iii) Find the union of the given sets Q and R
Solution:
(i) Union of sets P and Q is P ∪ Q
The smallest set which contains all the elements of set P and all the elements of set Q is {0, 2, 3, 4, 5, 6, 7,
9, 12}.
(ii) Union of two set P and R is P ∪ R
The smallest set which contains all the elements of set P and all the elements of set R is {2, 3, 4, 5, 6, 7,
8}.
(iii) Union of the given sets Q and R is Q ∪ R
The smallest set which contains all the elements of set Q and all the elements of set R is {0, 2, 3, 4, 6, 8,
9, 12}.

Notes:
A and B are the subsets of A ∪ B
The union of sets is commutative, i.e., A ∪ B = B ∪ A.
The operations are performed when the sets are expressed in roster form.
Some properties of the operation of union:
(i) A∪B = B∪A (Commutative law)
(ii) A∪(B∪C) = (A∪B)∪C (Associative law)
(iii) A ∪ ϕ = A (Law of identity element, is the identity of ∪)
(iv) A∪A = A (Idempotent law)
(v) U∪A = U (Law of ∪) ∪ is the universal set.
Notes:
A ∪ ϕ = ϕ ∪ A = A i.e. union of any set with the empty set is always the set itself.
Definition of Intersection of Sets:
Intersection of two given sets is the largest set which contains all the elements that are common to
both the sets.
To find the intersection of two given sets A and B is a set which consists of all the elements which
are common to both A and B.
The symbol for denoting intersection of sets is ‘∩‘.
For example:
Let set A = {2, 3, 4, 5, 6}
and set B = {3, 5, 7, 9}
In this two sets, the elements 3 and 5 are common. The set containing these common elements i.e.,
{3, 5} is the intersection of set A and B.
The symbol used for the intersection of two sets is ‘∩‘.

Therefore, symbolically, we write intersection of the two sets A and B is A ∩ B which means A
intersection B.
The intersection of two sets A and B is represented as A ∩ B = {x : x ∈ A and x ∈ B}
Solved examples to find intersection of two given sets:
1. If 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.
2. If X = {a, b, c} and Y = {ф}. Find intersection of two given sets X and Y.
Solution:
X ∩ Y = { }
3. If set A = {4, 6, 8, 10, 12}, set B = {3, 6, 9, 12, 15, 18} and set C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
(i) Find the intersection of sets A and B.
(ii) Find the intersection of two set B and C.
(iii) Find the intersection of the given sets A and C.
Solution:
(i) Intersection of sets A and B is A ∩ B
Set of all the elements which are common to both set A and set B is {6, 12}.
(ii) Intersection of two set B and C is B ∩ C
Set of all the elements which are common to both set B and set C is {3, 6, 9}.
(iii) Intersection of the given sets A and C is A ∩ C
Set of all the elements which are common to both set A and set C is {4, 6, 8, 10}.
Notes:
A ∩ B is a subset of A and B.
Intersection of a set is commutative, i.e., A ∩ B = B ∩ A.
Operations are performed when the set is expressed in the roster form.

Some properties of the operation of intersection
(i) A∩B = B∩A (Commutative law)
(ii) (A∩B)∩C = A∩ (B∩C) (Associative law)
(iii) ϕ ∩ A = ϕ (Law of ϕ)
(iv) U∩A = A (Law of ∪)
(v) A∩A = A (Idempotent law)
(vi) A∩(B∪C) = (A∩B) ∪ (A∩C) (Distributive law) Here ∩ distributes over ∪
Also, A∪(B∩C) = (AUB) ∩ (AUC) (Distributive law) Here ∪ distributes over ∩
Notes:
A ∩ ϕ = ϕ ∩ A = ϕ i.e. intersection of any set with the empty set is always the empty set.
How to find the difference of two sets?
If A and B are two sets, then their difference is given by A - B or B - A.
• If A = {2, 3, 4} and B = {4, 5, 6}
A - B means elements of A which are not the elements of B.
i.e., in the above example A - B = {2, 3}
In general, B - A = {x : x ∈ B, and x ∉ A}
• If A and B are disjoint sets, then A – B = A and B – A = B
Solved examples to find the difference of two sets:
1. A = {1, 2, 3} and B = {4, 5, 6}.
Find the difference between the two sets:
(i) A and B
(ii) B and A
Solution:
The two sets are disjoint as they do not have any elements in common.
(i) A - B = {1, 2, 3} = A

(ii) B - A = {4, 5, 6} = B
2. Let A = {a, b, c, d, e, f} and B = {b, d, f, g}.
Find the difference between the two sets:
(i) A and B
(ii) B and A
Solution:
(i) A - B = {a, c, e}
Therefore, the elements a, c, e belong to A but not to B
(ii) B - A = {g)
Therefore, the element g belongs to B but not A.
3. Given three sets P, Q and R such that:
P = {x : x is a natural number between 10 and 16},
Q = {y : y is a even number between 8 and 20} and
R = {7, 9, 11, 14, 18, 20}
(i) Find the difference of two sets P and Q
(ii) Find Q - R
(iii) Find R - P
(iv) Find Q – P
Solution:
According to the given statements:
P = {11, 12, 13, 14, 15}
Q = {10, 12, 14, 16, 18}
R = {7, 9, 11, 14, 18, 20}
(i) P – Q = {Those elements of set P which are not in set Q}

= {11, 13, 15}
(ii) Q – R = {Those elements of set Q not belonging to set R}
= {10, 12, 16}
(iii) R – P = {Those elements of set R which are not in set P}
= {7, 9, 18, 20}
(iv) Q – P = {Those elements of set Q not belonging to set P}
= {10, 16, 18}
In complement of a set if ξ be the universal set and A a subset of ξ, then the complement of A is the
set of all elements of ξ which are not the elements of A.
Symbolically, we denote the complement of A with respect to ξ as A’.
For Example; If ξ = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 7} find A'.
Solution:
We observe that 2, 4, 5, 6 are the only elements of ξ which do not belong to A.
Therefore, A' = {2, 4, 5, 6}
Note:
The complement of a universal set is an empty set.
The complement of an empty set is a universal set.
The set and its complement are disjoint sets.
For Example;
1. Let the set of natural numbers be the universal set and A is a set of even natural numbers,
then A' {x: x is a set of odd natural numbers}
2. Let ξ = The set of letters in the English alphabet.
A = The set of consonants in the English alphabet

then A' = The set of vowels in the English alphabet.
3. Show that;
(a) The complement of a universal set is an empty set.
Let ξ denote the universal set, then
ξ' = The set of those elements which are not in ξ.
= empty set = ϕ
Therefore, ξ = ϕ so the complement of a universal set is an empty set.
(b) A set and its complement are disjoint sets.
Let A be any set then A' = set of those elements of ξ which are not in A'.
Let x ∉ A, then x is an element of ξ not contained in A'
So x ∉ A'
Therefore, A and A' are disjoint sets.
Therefore, Set and its complement are disjoint sets
Similarly, in complement of a set when U be the universal set and A is a subset of U. Then the
complement of A is the set all elements of U which are not the elements of A.
Symbolically, we write A' to denote the complement of A with respect to U.
Thus, A' = {x : x ∈ U and x ∉ A}
Obviously A' = {U - A}
For Example; Let U = {2, 4, 6, 8, 10, 12, 14, 16}
A = {6, 10, 4, 16}
A' = {2, 8, 12, 14}
We observe that 2, 8, 12, 14 are the only elements of U which do not belong to A.
Some properties of complement sets
(i) A ∪ A' = A' ∪ A = ∪ (Complement law)
(ii) (A ∩ B') = ϕ (Complement law)

(iii) (A ∪ B) = A' ∩ B' (De Morgan’s law)
(iv) (A ∩ B)' = A' ∪ B' (De Morgan’s law)
(v) (A')' = A (Law of complementation)
(vi) ϕ' = ∪ (Law of empty set
(vii) ∪' = ϕ and universal set)
What is the cardinal number of a set?
The number of distinct elements in a finite set is called its cardinal number. It is denoted as n(A) and
read as ‘the number of elements of the set’.
For example:
(i) Set A = {2, 4, 5, 9, 15} has 5 elements.
Therefore, the cardinal number of set A = 5. So, it is denoted as n(A) = 5.
(ii) Set B = {w, x, y, z} has 4 elements.
Therefore, the cardinal number of set B = 4. So, it is denoted as n(B) = 4.
(iii) Set C = {Florida, New York, California} has 3 elements.
Therefore, the cardinal number of set C = 3. So, it is denoted as n(C) = 3.
(iv) Set D = {3, 3, 5, 6, 7, 7, 9} has 5 element.
Therefore, the cardinal number of set D = 5. So, it is denoted as n(D) = 5.
(v) Set E = { } has no element.
Therefore, the cardinal number of set D = 0. So, it is denoted as n(D) = 0.
Note:
(i) Cardinal number of an infinite set is not defined.
(ii) Cardinal number of empty set is 0 because it has no element.

Solved examples on Cardinal number of a set:
1. Write the cardinal number of each of the following sets:
(i) X = {letters in the word MALAYALAM}
(ii) Y = {5, 6, 6, 7, 11, 6, 13, 11, 8}
(iii) Z = {natural numbers between 20 and 50, which are divisible by 7}
Solution:
(i) Given, X = {letters in the word MALAYALAM}
Then, X = {M, A, L, Y}
Therefore, cardinal number of set X = 4, i.e., n(X) = 4
(ii) Given, Y = {5, 6, 6, 7, 11, 6, 13, 11, 8}
Then, Y = {5, 6, 7, 11, 13, 8}
Therefore, cardinal number of set Y = 6, i.e., n(Y) = 6
(iii) Given, Z = {natural numbers between 20 and 50, which are divisible by 7}
Then, Z = {21, 28, 35, 42, 49}
Therefore, cardinal number of set Z = 5, i.e., n(Z) = 5
2. Find the cardinal number of a set from each of the following:
(i) P = {x | x ∈ N and x22 < 30}
(ii) Q = {x | x is a factor of 20}
Solution:
(i) Given, P = {x | x ∈ N and x22 < 30}
Then, P = {1, 2, 3, 4, 5}
Therefore, cardinal number of set P = 5, i.e., n(P) = 5
(ii) Given, Q = {x | x is a factor of 20}

Then, Q = {1, 2, 4, 5, 10, 20}
Therefore, cardinal number of set Q = 6, i.e., n(Q) = 6
Cardinal Properties of Sets:
We have already learnt about the union, intersection and difference of sets. Now, we will go through
some practical problems on sets related to everyday life.
If A and B are finite sets, then
• n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
If A ∩ B = ф , then n(A ∪ B) = n(A) + n(B)
It is also clear from the Venn diagram that
• n(A - B) = n(A) - n(A ∩ B)
• n(B - A) = n(B) - n(A ∩ B)
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Problems on Cardinal Properties of Sets
1. If P and Q are two sets such that P ∪ Q has 40 elements, P has 22 elements and Q has 28 elements,
how many elements does P ∩ Q have?
Solution:

Given n(P ∪ Q) = 40, n(P) = 18, n(Q) = 22
We know that n(P U Q) = n(P) + n(Q) - n(P ∩ Q)
So, 40 = 22 + 28 - n(P ∩ Q)
40 = 50 - n(P ∩ Q)
Therefore, n(P ∩ Q) = 50 – 40
= 10
2. In a class of 40 students, 15 like to play cricket and football and 20 like to play cricket. How many
like to play football only but not cricket?
Solution:
Let C = Students who like cricket
F = Students who like football
C ∩ F = Students who like cricket and football both
C - F = Students who like cricket only
F - C = Students who like football only.
n(C) = 20 n(C ∩ F) = 15 n (C U F) = 40 n (F) = ?
n(C ∪ F) = n(C) + n(F) - n(C ∩ F)
40 = 20 + n(F) - 15
40 = 5 + n(F)
40 – 5 = n(F)
Therefore, n(F)= 35
Therefore, n(F - C) = n(F) - n (C ∩ F)
= 35 – 15
= 20
Therefore, Number of students who like football only but not cricket = 20
More problems on cardinal properties of sets

3. There is a group of 80 persons who can drive scooter or car or both. Out of these, 35 can drive
scooter and 60 can drive car. Find how many can drive both scooter and car? How many can drive
scooter only? How many can drive car only?
Solution:
Let S = {Persons who drive scooter}
C = {Persons who drive car}
Given, n(S ∪ C) = 80 n(S) = 35 n(C) = 60
Therefore, n(S ∪ C) = n(S) + n(C) - n(S ∩ C)
80 = 35 + 60 - n(S ∩ C)
80 = 95 - n(S ∩ C)
Therefore, n(S∩C) = 95 – 80 = 15
Therefore, 15 persons drive both scooter and car.
Therefore, the number of persons who drive a scooter only = n(S) - n(S ∩ C)
= 35 – 15
= 20
Also, the number of persons who drive car only = n(C) - n(S ∩ C)
= 60 - 15
= 45
4. It was found that out of 45 girls, 10 joined singing but not dancing and 24 joined singing. How
many joined dancing but not singing? How many joined both?
Solution:
Let S = {Girls who joined singing}
D = {Girls who joined dancing}
Number of girls who joined dancing but not singing = Total number of girls - Number of girls who
joined singing

45 – 24
= 21
Now, n(S - D) = 10 n(S) =24
Therefore, n(S - D) = n(S) - n(S ∩ D)
⇒ n(S ∩ D) = n(S) - n(S - D)
= 24 - 10
= 14
Therefore, number of girls who joined both singing and dancing is 14.
What are Venn Diagrams?
Pictorial representations of sets represented by closed figures are called set diagrams or Venn
diagrams.
Venn diagrams are used to illustrate various operations like union, intersection and difference.
We can express the relationship among sets through this in a more significant way.
In this,
• A rectangle is used to represent a universal set.
• Circles or ovals are used to represent other subsets of the universal set.
Venn diagrams in different situations
• If a set A is a subset of set B, then the circle representing set A is drawn inside the circle
representing set B.
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• If set A and set B have some elements in common, then to represent them, we draw two circles
which are overlapping.
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• If set A and set B are disjoint, then they are represented by two non-intersecting circles.
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In this diagrams, the universal set is represented by a rectangular region and its subsets by circles inside the
rectangle. We represented disjoint set by disjoint circles and intersecting sets by intersecting circles.
Multisets
A multiset is an unordered collection of elements, in which the multiplicity of an element may be
one or more than one or zero. The multiplicity of an element is the number of times the element
repeated in the multiset. In other words, we can say that an element can appear any number of
times in a set.
Example:
1. A = {l, l, m, m, n, n, n, n}
2. B = {a, a, a, a, a, c}
Operations on Multisets

1. Union of Multisets: The Union of two multisets A and B is a multiset such that the
multiplicity of an element is equal to the maximum of the multiplicity of an element in A and B
and is denoted by A ∪ B.
Example:
1. Let A = {l, l, m, m, n, n, n, n}
2. B = {l, m, m, m, n},
3. A ∪ B = {l, l, m, m, m, n, n, n, n}
2. Intersections of Multisets: The intersection of two multisets A and B, is a multiset such that
the multiplicity of an element is equal to the minimum of the multiplicity of an element in A and
B and is denoted by A ∩ B.
Example:
1. Let A = {l, l, m, n, p, q, q, r}
2. B = {l, m, m, p, q, r, r, r, r}
3. A ∩ B = {l, m, p, q, r}.
3. Difference of Multisets: The difference of two multisets A and B, is a multiset such that the
multiplicity of an element is equal to the multiplicity of the element in A minus the multiplicity
of the element in B if the difference is +ve, and is equal to 0 if the difference is 0 or negative
Example:
1. Let A = {l, m, m, m, n, n, n, p, p, p}
2. B = {l, m, m, m, n, r, r, r}
3. A - B = {n, n, p, p, p}
4. Sum of Multisets: The sum of two multisets A and B, is a multiset such that the multiplicity
of an element is equal to the sum of the multiplicity of an element in A and B
Example
1. Let A = {l, m, n, p, r}
2. B = {l, l, m, n, n, n, p, r, r}
3. A + B = {l, l, l, m, m, n, n, n, n, p, p, r, r, r}

5. Cardinality of Sets: The cardinality of a multiset is the number of distinct elements in a
multiset without considering the multiplicity of an element
Example:
1. A = {l, l, m, m, n, n, n, p, p, p, p, q, q, q}
The cardinality of the multiset A is 5.
Ordered Set
It is defined as the ordered collection of distinct objects.
Example:
1. Roll no {3, 6, 7, 8, 9}
2. Week Days {S, M, T, W, W, TH, F, S, S}
Ordered Pairs
An Ordered Pair consists of two elements such that one of them is designated as the first member
and other as the second member.
(a, b) and (b, a) are two different ordered pair. An ordered triple can also be written regarding an
ordered pair as {(a, b) c}
An ordered Quadrable is an ordered pair {(((a, b), c) d)} with the first element as ordered triple.
An ordered n-tuple is an ordered pair where the first component is an ordered (n - 1) tuples, and
the nth
element is the second component.
1. {(n -1), n}
Example:

SETS in Discrete Structure

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