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THEORY
OF
SETSTeam members
1. Ridah Tarannum Mehmood
2. Nanjiba Ibnat Shahid
3. Md Nahian Hasan
4. Taiseer Ahmed
CONTENTS
•BIRTH OF SET THEORY
•DEFINITION
•EXAMPLE OF SETS
•CLASSIFICATION OF SET
•SYMBOLS
•OPERATIONS
•LAWS
•SET IN DAILY LIFE
• Between the years 1874 and 1897, the German mathematician
Georg Cantor formulated a theory of abstract sets of entities and
transformed it into a mathematical discipline- Set.
• This theory came out from his thorough investigations of some
concrete problems regarding certain types of infinite sets of real
numbers.
• According to Cantor, a set is a collection of definite,
distinguishable objects of perception or thought conceived as a
whole.
• The objects are called elements of the set.
George Cantor
1845-1918
• A set is any collection of objects specified in such a
way that we can determine whether a given object is
or is not in the collection.
• The symbol ‘∈’ is used to denote an
element of a set.
• Elements in a set do not follow any
order.
Brief Concept on Set Theory
CLASSIFICATION OF
SET
1.Null set
2.Singleton set
3.Finite and
Infinite set
4.Equivalent
and Equal
set
5.Universal set
6.Subset
7.Power set
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 {}.
Example:
• { x:x is real number and x2<0} is an empty
set as the square of a real number is always positive.
A B
A∩B = Null set
Singleton
set
A set which contains only one element is
called a singleton set.
Examples
B={ x:x is an even prime number}
So, B={2} as 2 is the only even prime number
2
B
Finite and
Infinite Set
A set which contains a definite number of
elements is called a finite set. Also , empty
set is called finite set.
N = {x : x ∈ N, x < 7}
P = {2, 3, 5, 7, 11, 13, 17, ...... 97}
The set whose elements cannot be listed, i.e.,
set containing never-ending elements is
called an infinite set.
A = {x : x ∈ N, x > 1}
Set of all prime numbers
B = {x : x ∈ W, x = 2n}
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 ‘↔’.
4
5
6
d
e
f
A={4,5,6} B={d,e,f}
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.
Equivalent Set Equal Set
1
2
3
2
3
1
A B
A universal set is a set which contains all
objects, including itself.
Example:
A={ x:x is a positive integer and 5x<16}
B={ x:x is a positive integer and x2<20}
U={ x:x is a positive integer}
Universal Set
A power set is the set of all subsets of a particular
set
Example:
Let A={1,2} be a set. Then the subsets of A are
{1}, {2}, {1,2}, {}.
So, the power set of A, P(A)= { {}, {1}, {2}, {1,2}
}.
The number of elements in a power set of a set can
be find by the below formula:-
2n ; here “n” denotes the number of elements in
the original set.
Power Set
Subset
B
A
A is a proper subset
of B, A⊂B, and
conversely B is a
proper superset of
A.
• If A and B are sets and every element of A is also an
element of B, then
• A is a subset of B, denoted by 𝐴 ⊆ 𝐵, or equivalently
• B is a superset of A, denoted by 𝐵 ⊇ 𝐴
• If A is a subset of B, but A is not equal to B then,
• A is a proper (or strict) subset of B, denoted by 𝐴 ⊊ 𝐵, 𝑜𝑟
equivalently
• B is a proper (or strict) superset of A, denoted by 𝐵 ⊋ 𝐴
SYMBOLS SHORT NOTE
• Arrangement of the
elements does not
have any effect in a set
• Repetition of elements
does not have any
effect in a set.
• “∈” this symbol means an element of/belongs
to.
Example:
Let A={1,2} be a set. Then 2 ∈ A which means 2
belongs to A.
• “∉” this symbol means “not an element”.
Example:
Let B={ 3,4} be a set. Then 5 ∉ B.
Operations
In set, the union (denoted by ∪) of a
collection of sets is the set of
all elements in the collection. It is one of
the fundamental operations through
which sets can be combined and related to
each other.
Example:
A={ 1,2,3} , B={2,4,6}
A ∪ B = {1,2,3} ∪ {4,2,6}
i.e., A ∪ B = {1,2,3,4,6}
Union = ∪
B
A
4
3
1
2
6
Intersection =∩
Intersect of set is the set which contains
the common elements of two or more
sets.
Example:
A = {1,2,3}, B = {4,2,6}
A ∩ B = {1,2,3} ∩ {4,2,6}
= {2}
B
A
4
3
1
2
6
Difference =(–),()
If A and B are sets, then the relative
complement of A in B, also termed the set
difference of A and B, is the set of
elements in A but not in B.
A={1,2, 3, 4}, B={2, 4, 5}
A-B={1, 2, 3, 4} – { 2, 4, 5}
={ 1,3}
B
A
4
3
1 2 5
Complement =A
c
The complement of set A is defined as a set that
contains the elements present in the universal set
but not in set A .
A={1, 2, 3}
U={1, 2, 3, 4, 5, 6}
AC= U-A
={4, 5, 6}
4
A
1,2,3
4 5
6 U
Operations
Laws of Set
Idempotency
•AUA=A
•A∩A=A
Associative
laws
• 𝑨 ∪ 𝑩 ∪ 𝑪 =
𝑨 ∪ 𝑩 ∪ 𝑪
• 𝑨 ∩ 𝑩 ∩ 𝑪 =
𝑨 ∩ (𝑩 ∩ 𝑪)
Distributive
laws
• 𝑨 ∪ (𝑩 ∩
𝑪)= 𝑨 ∪ 𝑩 ∩
𝑨 ∪ 𝑪
• 𝑨 ∩ (𝑩 ∪ 𝑪) =
(𝑨 ∩ 𝑩) ∪ (𝑨 ∩
𝑪)
Laws of Set
De-Morgan’s laws
• (𝐴 ∪ 𝐵)/
= 𝐴/
∩ 𝐵/
• (𝐴 ∩ 𝐵)/
= 𝐴/
∪ 𝐵/
Commutative laws
•A∪B = B∪A
•A∩B = B∩A
Set in Daily Life
• Kitchen is the most relevant
example of sets. The kitchen is
always well arranged. The
plates are kept separate from
bowls and cups. Sets of similar
utensils are kept separately
Kitchen
• School bags are also an
example. There are
usually divisions in the
school bags, where the
sets of notebooks and
textbooks are kept
separately
School Bags
• When we go shopping in a
mall, we all have noticed that
there are separate portions
for each kind of things. For
instances, clothing shops are
on another floor whereas
the food court is at
another part of the mall
Shopping Mall
Set in Daily Life
• Most of us have a different
kind of playlists of songs
present in our smartphones
and computers. Rock songs
are often separated from
classical or any other genre.
Hence, playlists also form
the example of sets.
Playlist
• Corporate offices are
examples of sets. Here the
people belonging to various
departments have to sit
separately from other
departments.
Office
• Computer Science
• Data Structure
• Topology
• Physics
• Computational Economics
Different
fields
THANK YOU

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Brief Concept on Set Theory

  • 1. THEORY OF SETSTeam members 1. Ridah Tarannum Mehmood 2. Nanjiba Ibnat Shahid 3. Md Nahian Hasan 4. Taiseer Ahmed
  • 2. CONTENTS •BIRTH OF SET THEORY •DEFINITION •EXAMPLE OF SETS •CLASSIFICATION OF SET •SYMBOLS •OPERATIONS •LAWS •SET IN DAILY LIFE
  • 3. • Between the years 1874 and 1897, the German mathematician Georg Cantor formulated a theory of abstract sets of entities and transformed it into a mathematical discipline- Set. • This theory came out from his thorough investigations of some concrete problems regarding certain types of infinite sets of real numbers. • According to Cantor, a set is a collection of definite, distinguishable objects of perception or thought conceived as a whole. • The objects are called elements of the set. George Cantor 1845-1918
  • 4. • A set is any collection of objects specified in such a way that we can determine whether a given object is or is not in the collection. • The symbol ‘∈’ is used to denote an element of a set. • Elements in a set do not follow any order.
  • 6. CLASSIFICATION OF SET 1.Null set 2.Singleton set 3.Finite and Infinite set 4.Equivalent and Equal set 5.Universal set 6.Subset 7.Power set
  • 7. 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 {}. Example: • { x:x is real number and x2<0} is an empty set as the square of a real number is always positive. A B A∩B = Null set
  • 8. Singleton set A set which contains only one element is called a singleton set. Examples B={ x:x is an even prime number} So, B={2} as 2 is the only even prime number 2 B
  • 9. Finite and Infinite Set A set which contains a definite number of elements is called a finite set. Also , empty set is called finite set. N = {x : x ∈ N, x < 7} P = {2, 3, 5, 7, 11, 13, 17, ...... 97} The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set. A = {x : x ∈ N, x > 1} Set of all prime numbers B = {x : x ∈ W, x = 2n}
  • 10. 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 ‘↔’. 4 5 6 d e f A={4,5,6} B={d,e,f} 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. Equivalent Set Equal Set 1 2 3 2 3 1 A B
  • 11. A universal set is a set which contains all objects, including itself. Example: A={ x:x is a positive integer and 5x<16} B={ x:x is a positive integer and x2<20} U={ x:x is a positive integer} Universal Set A power set is the set of all subsets of a particular set Example: Let A={1,2} be a set. Then the subsets of A are {1}, {2}, {1,2}, {}. So, the power set of A, P(A)= { {}, {1}, {2}, {1,2} }. The number of elements in a power set of a set can be find by the below formula:- 2n ; here “n” denotes the number of elements in the original set. Power Set
  • 12. Subset B A A is a proper subset of B, A⊂B, and conversely B is a proper superset of A. • If A and B are sets and every element of A is also an element of B, then • A is a subset of B, denoted by 𝐴 ⊆ 𝐵, or equivalently • B is a superset of A, denoted by 𝐵 ⊇ 𝐴 • If A is a subset of B, but A is not equal to B then, • A is a proper (or strict) subset of B, denoted by 𝐴 ⊊ 𝐵, 𝑜𝑟 equivalently • B is a proper (or strict) superset of A, denoted by 𝐵 ⊋ 𝐴
  • 13. SYMBOLS SHORT NOTE • Arrangement of the elements does not have any effect in a set • Repetition of elements does not have any effect in a set. • “∈” this symbol means an element of/belongs to. Example: Let A={1,2} be a set. Then 2 ∈ A which means 2 belongs to A. • “∉” this symbol means “not an element”. Example: Let B={ 3,4} be a set. Then 5 ∉ B.
  • 14. Operations In set, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. Example: A={ 1,2,3} , B={2,4,6} A ∪ B = {1,2,3} ∪ {4,2,6} i.e., A ∪ B = {1,2,3,4,6} Union = ∪ B A 4 3 1 2 6 Intersection =∩ Intersect of set is the set which contains the common elements of two or more sets. Example: A = {1,2,3}, B = {4,2,6} A ∩ B = {1,2,3} ∩ {4,2,6} = {2} B A 4 3 1 2 6
  • 15. Difference =(–),() If A and B are sets, then the relative complement of A in B, also termed the set difference of A and B, is the set of elements in A but not in B. A={1,2, 3, 4}, B={2, 4, 5} A-B={1, 2, 3, 4} – { 2, 4, 5} ={ 1,3} B A 4 3 1 2 5 Complement =A c The complement of set A is defined as a set that contains the elements present in the universal set but not in set A . A={1, 2, 3} U={1, 2, 3, 4, 5, 6} AC= U-A ={4, 5, 6} 4 A 1,2,3 4 5 6 U Operations
  • 16. Laws of Set Idempotency •AUA=A •A∩A=A Associative laws • 𝑨 ∪ 𝑩 ∪ 𝑪 = 𝑨 ∪ 𝑩 ∪ 𝑪 • 𝑨 ∩ 𝑩 ∩ 𝑪 = 𝑨 ∩ (𝑩 ∩ 𝑪) Distributive laws • 𝑨 ∪ (𝑩 ∩ 𝑪)= 𝑨 ∪ 𝑩 ∩ 𝑨 ∪ 𝑪 • 𝑨 ∩ (𝑩 ∪ 𝑪) = (𝑨 ∩ 𝑩) ∪ (𝑨 ∩ 𝑪)
  • 17. Laws of Set De-Morgan’s laws • (𝐴 ∪ 𝐵)/ = 𝐴/ ∩ 𝐵/ • (𝐴 ∩ 𝐵)/ = 𝐴/ ∪ 𝐵/ Commutative laws •A∪B = B∪A •A∩B = B∩A
  • 18. Set in Daily Life • Kitchen is the most relevant example of sets. The kitchen is always well arranged. The plates are kept separate from bowls and cups. Sets of similar utensils are kept separately Kitchen • School bags are also an example. There are usually divisions in the school bags, where the sets of notebooks and textbooks are kept separately School Bags • When we go shopping in a mall, we all have noticed that there are separate portions for each kind of things. For instances, clothing shops are on another floor whereas the food court is at another part of the mall Shopping Mall
  • 19. Set in Daily Life • Most of us have a different kind of playlists of songs present in our smartphones and computers. Rock songs are often separated from classical or any other genre. Hence, playlists also form the example of sets. Playlist • Corporate offices are examples of sets. Here the people belonging to various departments have to sit separately from other departments. Office • Computer Science • Data Structure • Topology • Physics • Computational Economics Different fields