Tuesday, April 2, 2024

Set Theory Introduction | Definition Of Sets

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set theory introduction
 introduction to set theory

Definition of set: (set theory introduction)

A Set is the group of well Defined elements or the object.
For example set of furniture object, set of any particular class student, set of vegetables etc.
A set is always denoted by the capital letter of English letter.

Example of the set:

Let B={1,3,5,7,9} be any arbitrary set thus the set has elements 1,3,5,7,9 . It can understand by that 1∈B , 3∈B , 5∈B ,7∈B and 7∈B
Here are the few example of the commonly used set which are used Commonly in in lower and upper class mathematics.

Set of Natural number ={1,2,3,4,5,6,7,..........} denoted by .
Set of the Real number ={........-3,-3/2,-1,0,1,4/2,6/2.5,3.5,6,...........} denoted by  R.
Set of Integers = {..........-5,-4,-3,-2,-1,0,1,2,3,4,5,.................} denoted by Z.
Set of rational numbers (which are in p/q format where q≠0) ={......-6/2,-5/2,-2/2,0/1,3/3,4/2,6/2,......} denoted by Q .

Cardinality/Order of the Set:

Cardinality Of any set is known as the total number of the element in that set. so cardinality basically describe the size of any set. according to the We can classify to the set into Finite and infinite.

Classification of Sets:

There are many sets in Existance like finite set, infinite set, empty set, equal set, singleton set etc;

Finite Set:

A Set which contains finite numer of elements known as finite set.
For Example Set of positive integers upto 5.

Infinite Set :

A Set which does not contains finite numer of elements known as Infinite set.
For Example Set of integers.

Empty Set :

A Set which does not contains any elements known as Empty set. Its also known as Null set. Its denoted by { } or Ø.

Equivalent/Equal Set :

Its applied for two sets; in which the cardinality for both sets will same known as the equal sets . 
for example let A={2,4,6} and B={1,3,5} then both the sets will become equal.

Singleton Set:

A set which contains only one element known as Singleton set.
Example A={1}

Disjoint Sets:

Its also applied for two sets but both the two sets will differ to each other in other word two Set A and B are known as disjoint if there is not any single element in common.
For Example let A={2,4,6,8} and B={1,3,5,7}


The set B is known to be a subset of A if every element of B is also a member of B.
Its denoted as B ⊆ A; Empty set is also the subset of another set. generally a subset is a part of another set.
For Example: A = {1,2,3}  , B={2,3}

Then B ⊆ A.

Possible Subsets of set A are: {},{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}.It means {} is subset of every sets and

Every set is also a subset of itself  ( if A is any set then A⊆A )
If B is not subset of A then its denoted by B⊄A.


The Set B is known as the superset of Set A if all the elements of set A are the elements of set B.
Its denoted as B ⊃ A.

For example Let B = {0,1,2,3,4,5} and set A = {0, 3, 4}; then set B is the superset of Set A.

Proper Subset:

If two sets such that  A ⊆ B and A ≠ B  then A is known as proper subset of B and its represented as A⊂B.

Example: If A = {1,2,3} is  a subset of B = {1,2,3,4,5} then Set A is the proper subset of set B.

Operations on Sets:

Set operation is done between two or among more than two sets; few basic operation on the sets are as follows;

Union of Sets:

Let there are two set known as set A and set B  then A union B is the set which contains all the elements of both the set A and set B. Its represented as A ∪ B.

Example: Set A = {0,1,2,3} and B = {2,5,6};

then A Union B ;   A ∪ B = {0,1,2,3,5,6}

Intersection of Sets:

Let there are two set known as set A and set B then A intersection B is the set that contains only the common elements between set A and set B;
Its represented as A ∩ B.

Example: Set A = {0,1,2,3}  and B = {0,1,6}, 
then A intersection B is ; A ∩ B = {0,1}

If there is no common element between two or more than two sets then their intersection will be Null.
Eg. let A ={1,2,3} and B={4,5,6} then A intersection B i.e.  A ∩ B =  { } or Ø

Complement of Sets:

Let A be any set then complement of set A is the set of all elements in the universal set which is not the element of A . It is denoted by A’.

For example Let A = {1,2,3 } let universal set U = {1,2,3,4,5,6,7,8,9 } then complement of set A
 i.e. A’ = {4,5,6,7,8,9} 

Properties of Complement Sets:

  • A ∪ A′ = U
  • A ∩ A′ = Φ
  • Law of double complement : (A′ )′ = A that is complement of the complement set is equal to the set itself.
  •   Φ′ = U and U′ = Φ.

Difference of Sets:

Let Set A and Set B be any two sets then their A difference B means set of elements which is the element of set A but not set B. Its denoted as A-B.

For Example A={1,2,3,4} and B= {1,2} ;
then A-B ={3,4}

Cartesian Product of sets: 

Cartesian of two set are the ordered pair element such as first elements belongs to first set and second belongs to second in case of two set

For example Let A and B be any two set then A x B is the ordered pair of (a ,b) such that a ∈A and b∈ B
let A ={1,2}  and B ={ 3,4} then A x B =  { (1,3),(1,4),(2,3),(2,4) }

For any three sets A, B and C some important formulas are follows;
  • n ( A ∪ B ) = n(A) + n(B) – n ( A ∩ B)
  • n( A – B) + n( A ∩ B ) = n(A)
  • n( B – A) + n( A ∩ B ) = n(B)
  • If A ∩ B = ∅, then n ( A ∪ B ) = n(A) + n(B)
  • n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B )
  • n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) +  n ( A ∩ B  ∩ C)
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