Super Set : Let A and B be two sets . If A ⊂ B and A ≠ B , B is called superset of A .
1 . The set Q of rational numbers is a subset of the set R of real numbers . We write Q ⊂ R
2 . Let A = { 1 , 3 , 5 } and B = { x : x is an odd natural number less than 6 }. Then A ⊂ B and B ⊂ A and hence A = B .
3 . Let A = { a , e , i , o , u } and B = { a , b , c , d }. Then A is not a subset of B , also B is not a subset of A .
4 . Let A ={ 1,2,3,4 } and B ={ 1,2,3,4,5,6 }, then A is subset of B , and B is super set of A .
5 . Some relation in well defined sets : N ⊂ Z ⊂ Q ⊂ R
N :{ 1 , 2 , 3 , 4 , 5 …} Z : ( -7, -6, -5, 1 , 4 , 5 , ..} Q : { 1.2 , 1.3 , 1.5 , 2.2 ..} R : { pie , 1 , 3 ..}
Natural number Integers Rational Numbers Real Number
Singleton Set If a set A has only one element , we call it a singleton set . Thus { a } is a singleton set . E . g . C ={ x : x ∈ N + and x 2 = 4 } , it has only one element C ={ 2 }
Power Set The collection of all subsets of a set A is called the power set of A .
E . g . Consider the set { 1 , 2 }. Let us write down all the subsets of the set { 1 , 2 }. Subsets of { 1,2 } are : φ , { 1 }, { 2 } and { 1 , 2 }.
The set of all these subsets is called the power set of { 1 , 2 }.
In general , if A is a set with n ( A ) = m , then it can be shown that n [ P ( A )] = 2 m