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Sets , Cardinality and Power Set in Set Theory

Sets , Cardinality  and Power Set in Set Theory

Set - Set is an ordered collection of distinct elements . Set is the part of basically Discrete Mathematics but widely used in most of the fields as Set Thory , Permutations and Combinations , Artificial Intelligence , Machine language etc in the form of Sets .

example -
A = { 1 , 2 , 3 , 4 , 5}
B = { 1 , 2 , 2 ,3 ,4}
a = { 1 , 2 , 3 , 4 , 5}
C = { 1 , 2 , A , B , क , ख  , ग }

Note -

  • Here A is a Set because it contains all the distinct elements .
  • Here B is not a Set because it contains duplicate element 2 .
  • a is not a Set because it must be represented by capital letter as A .
  • C is a Set if all the letters of Hindi , english , symbols , Integers are allowed Otherwise not a Set .  
Cardinality of Set - It shows the number of elements available in that Set .

example - 
A = { 1,2,3,4,5}
Cardinality of A = | A | = 5 

Representation of Sets - There are several methods to represent a set like - 

1. Tabular / Roaster Method - This method basically applies on the sets of small number of elements that are possible to show as a listing or as a table .

example - 
A = { 1,2,3,4,5 }
B = { ......-3 , -2 , -1 , 0 , 1 , 2 , 3 ........n }
C = { a , b , c , d }

2. Set builder Representation - This method basically applies on the sets of large number of elements .

example - 
A = { x | x ε N  &&  x < 6 }
B = { 2x | x ε Z }

Types of Sets - 

Types of Sets


Various Number Sets -

Number Set Range
Natural Number ( N )  { 1 , 2 , 3 , 4, 5 .......∞}
Integers ( Z )  { -∞ ......-2 , -1 , 0 , 1 , 2 ........ +∞ }
Real Numbers ( R )  { -∞ to +∞ }
Whole Numbers ( W )  { 0 , 1 ,2 ,3 ,  4 , 5 .......+∞ }

Subset - 
  • A is always a subset of A means A ⊆ A .
  • If A ⊆ B and B ⊆ A then A = B .
  • All the Subsets are parts of Sets and Set is a part of Universal Set ( U ) .
  • Empty Set is just opposite of Empty Set ( Φ ) . Empty set does not contain any elements inside it .
Power Set - Let A be a Finite Set , then Set of all Subsets of Set A is called Power Set .

example - 
If A = { 1 , 2 , 3 }
then P( A ) = { Φ , {1} , {2} , {3} , {1,2} , {1,3} , {2,3} , {1,2,3} }

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