What is the total number of proper subsets of a set containing n elements?

Explanation

Let P(n) : number of subset of a set containing n distinct elements `2^(n)`, for all n `in N`.
Step I We observe that of P(1) is true, for n=1.
Number of subsets of a set contain 1 element is `2^(1)=2`, which is true.
Step II Assume that P(n) is true for n=k.
P(k) : Number of subsets of a set containing k distinct elements is `2^(k)`, which is true.
Step III To prove P(k+1) is true, we have to show that
P(k+1): Number of subsets of a set containing (k+1) distinct elements is `2^(k+1)`
We know that, with addition of one element in the set number of subsets become double.
`:.` Number of subsets of a set containing (k+1) distinct elements `=2xx2^(k)=2^(k+1)`.
So, P(k+1) is true. Hence, P(n) is true.