This document defines and provides examples of sets. The key points covered are:
- A set is a collection of well-defined objects. Sets can be represented using curly braces and commas to separate elements.
- There are different types of sets including finite sets with a countable number of elements, infinite sets with an unlimited number of elements, empty/null sets with no elements, and singleton sets containing one element.
- Set operations like union, intersection, and complement are introduced along with their properties and examples. A Venn diagram can show the relationships between sets visually.
Know the basics on sets such as the methods of writing sets, the cardinality of a set, null and universal sets, equal and equivalents sets, and many more.
Know the basics on sets such as the methods of writing sets, the cardinality of a set, null and universal sets, equal and equivalents sets, and many more.
How To Write Set Notation for set properties such as:
Union - Elements in Set A or Set B
Intersection - Elements in Set A and Set B
Cardinality - the count of elements within the set
Elements - Items within set. Can be numeric or objects
Empty Set - The set of nothing
Such That - Defining which elements belong to a set
Universal Set - The set of all possible values
Complement - Everything not in the set in the Universal Set
Subset - A subset is a set of elements which are all an element of another set
Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
After going through this module, you are expected to:
• define well-defined sets and other terms associated to sets
• write a set in two different forms;
• determine the cardinality of a set;
• enumerate the different subsets of a set;
• distinguish finite from infinite sets; equal sets from equivalent sets
• determine the union, intersection of sets and the difference of two sets
How To Write Set Notation for set properties such as:
Union - Elements in Set A or Set B
Intersection - Elements in Set A and Set B
Cardinality - the count of elements within the set
Elements - Items within set. Can be numeric or objects
Empty Set - The set of nothing
Such That - Defining which elements belong to a set
Universal Set - The set of all possible values
Complement - Everything not in the set in the Universal Set
Subset - A subset is a set of elements which are all an element of another set
Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
After going through this module, you are expected to:
• define well-defined sets and other terms associated to sets
• write a set in two different forms;
• determine the cardinality of a set;
• enumerate the different subsets of a set;
• distinguish finite from infinite sets; equal sets from equivalent sets
• determine the union, intersection of sets and the difference of two sets
Set theory is a monumental concept in the world of mathematics. Starting from business to even literature, set has uses in diverse fields. This pdf presents set in a unique and eye-catching way. Hope you guys enjoy it.
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3. Examples
•First 10 counting numbers 1, 2, ………. 10
• Brave children in a class
• Planets in our solar system M V E M J S U N
• Days of the week Sunday, Monday, …….. Saturday
• Months in a year January, February, ……….. December
•Strong forts of Maharashtra
The
collection
of well
defined
objects is
called
set.
4. Representation of Set
A = { a, e, i, o, u }
• Use curly braces
• Do not repeat the elements
• Use comma to separate the members of the set.
Name of the
set : Capital
letter
Members/Elements of the set :
Small letters
5. •Consider
•A = { a, e, i, o, u }
• a Є A a is a member of set A OR a belongs to set A
OR a is the element of set A
• b Є A b is not a member of set A OR b does not
belongs to set A OR b is not a element of set A
6. Methods of writing sets
1. Listing or Roster
method
• all the elements are
listed and are separated
by a comma.
2. Rule method or Set
builder form
• the elements of the set
are represented by a
variable followed by a
vertical line or colon and
the property of the
variable is defined.
7. Examples
Sr.
No.
Listing Method Rule Method
1. X={Sunday, Monday, Tuesday,
Wednesday, Thursday, Friday,
Saturday}
X={x | x is a day of a week}
2. A={0,1,2,3,…….20} A={x | xЄW , x<21}
3. B={I, N, D, A} B={y | y is a letter of the word ‘INDIA’}
4. D={3,6,9,12,15,18,…….} D={x | x is a multiple of 3}
9. 1. Singleton Set
•A set containing only one element is called Singleton Set
•A = {x | x is even prime number}
• A={2}
•B = {x | x is neither prime nor composite}
• B={1}
•C = {x | x is the smallest natural number}
• C={1}
•D = {x | x is the smallest whole number}
• D={0}
10. 2. Empty or Null Set
•If there is not a single element in the set which satisfies the
given condition then it is called a Null set or Empty set.
•If a set does not contain any element then it is called a Null
set or Empty set.
•A = {x | x is a natural number between 2 and 3}
• A={ } OR A=Ф
•B = {x | x Є N, x<1}
• B={ }
•C = {x | x is prime number, x<2}
• C={ }
11. 3. Finite Set
• If a set contains countable number of elements then the set
is called Finite set.
• If a set is a null set or the number of elements are limited
then the set is called Finite set.
• A = { a, e, i, o, u}
• B = {1, 2, 3, 4, 5, 6, 7}
• C = {x | x Є W, x < 3}
• C = {0, 1, 2}
• D = {y | y is a prime number, y < 20}
• D = {2, 3, 5, 7, 11, 13, 17, 19}
12. 4. Infinite Set
• If the number of elements in a set are unlimited or
uncountable then the set is called Infinite set.
• N = {1, 2, 3, 4, ………}
• A = {x | x is a multiple of 2}
• A={2, 4, 6, 8, ………}
•B= {y | y is an odd number}
• B = {1, 3, 5, 7, ………..}
N, W, I, Q, R all these sets are Infinite sets.
13. Equal Sets
•Two sets A and B are said to be equal
•If all the elements of set A are present in set B AND all the
elements of set B are present in set A.
•It is represented as A = B
•A = {x | x is a letter of the word ‘listen’} A = {l, i, s, t, e, n}
•B = {x | x is a letter of the word ‘silent’} B = {s, i, l, e, n, t}
•A = B
•C = {y | y is a prime number, 2 < y < 9} C = {3, 5, 7}
•D = {y | y is an odd number, 1 < y < 8} D = {3, 5, 7}
•C = D
14. Venn Diagrams
( British logician ) was the first to use
closed figures for representing a set.
• Venn diagrams help us to understand the
relationship among sets.
• Eg: A = { 1, 2, 3, 4, 5 }
A
1
2
3
4
5
15. Subset
•It is written as A C B.
•It is read as “A is a subset of B”
or “A subset B”.
•Eg: B = { 1, 2, 3, 4, 5 }
A = { 1, 2, 3 }
16. Points to remember
1) Every set is a subset of itself. -> A C A
2) Empty set is a subset of every set. -> ɸ C A
Consider:
A = { 2, 4, 6, 8 } and B = { 2, 4, 6, 8 }
3) If A = B then A C B and B C A
4) If A C B and B C A then A = B
17. Universal Set
•A set which can accomodate
all the given sets under
consideration is known as
.
•It is generally denoted by 'U'.
•In Venn diagram it is denoted
by a rectangle.
18. Complement of a set
• A complement of a set is
the set of those
elements which does not
belong to the given set
but belongs to the
universal set.
19. Properties of complement of a set
• No elements are common in A and A'.
• A C U and A' C U
• Complement of set U is an empty set. U' = ɸ
• Complement of empty set is U. ɸ' = U
21. Intersection of two sets
• The intersection of sets
A and B is the set that
contains the common
elements of set A and
set B.
22. Properties of Intersection of sets
1) A ∩ B = B ∩ A
3) If A ∩ B = B then B C A
A = {1, 3, 2 } & B = {1, 2}
A ∩ B = { 1, 2 } = B
5) A ∩ A' = ɸ
U = { T, S, U, N, A, M, I }
A = { S, U, N} ; A' = {T, A, M, I}
6) A ∩ A =A
7) A ∩ ɸ = ɸ
24. Union of two sets
• The union of sets A and B
is the set that contains all
the elements of both the
sets.
• It is written as A U B.
• It is read as “A union B”
25. Properties of Union of sets
1) A U B = B U A 4) A U A' = U
U = { T, S, U, N, A, M, I }
A = { S, U, N} ; A' = {T, A, M, I}
26. Number of elements in a set
•A = { 2, 4, 6, 8, 10 } & B = { 1, 2, 6, 9 }
•n(A) = 5 & n(B) = 4 n(A) + n(B) = 9
•A U B = { 1, 2, 4, 6, 8, 9, 10 } A∩B = { 2, 6 }
n(A U B) + n(A∩B) =9
•means n(A) + n(B) = n(A U B) + n(A∩B)