Brief Concept on Set Theory

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.

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

Brief Concept on Set Theory

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