Question

If a and b are non empty set such that a proper subset of b then?

Answer 1

Answer:

The set of all subsets of a A is the power set of A. The whole set (A itself) and the empty set (null set),ϕ are two of the elements in the power set. While the empty set and all other elements (except the whole set A) of the power set of A are proper subsets of A, the whole set A cannot be said to be a proper subset of itself.

The cardinality of the power set of A is 2 raised to the cardinality of A.

After denoting the cardinality of A, the power set of A, and the cardinality of the power set of A by n(A), P(A) and n(P(A)) respectively, then by the definition of the cardinality of the power set we have

n(P(A))=2n(A)

From the question, the number of proper subsets is 15. This added to just 1 element, the whole set, gives the cardinality of the power set, P(A)

By that,

n(P(A))=15+1=16

Combining with the first equation,

2n(A)=16=24

This implies

n(A)=4

The required cardinality is 4