Find the number of a proper subset of a set whose cardinal number is $4$.

Show with an example that if the number of elements in a set is $n$ then the number of subsets is $2^n$.

Find the number of a proper subset of a set whose cardinal number is $4$.

If two sets $A \& B$ are $A=\left\{1,2, 3, 4\right\} and B=\left\{2, 3, 4, 5\right\},$ then what is the set containing elements which are either in $A or in B$ or in both?

The number of non-empty subsets of a given set $A$ is $63$. Find the cardinal number of set $A$.

Two finite sets have $p \& q$ number of elements respectively. If the total number of subsets of the first set is $24$ more than the total number of subsets of the second set, then find the values of $p \& q$. 

If the power set of a set has $32$ elements, then how many elements does the set have? Name the set which is not a subset of any of the given sets.

Cardinal number of a set can never be negative.

If two sets have same cardinal number, then they are equal sets.