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