Prove that the total number of subsets of a finite set containing n elements is \(2^n\).

Solution :

Let A be a finite set containing n elements. Let 0 \(\le\) r \(\le\) n.

Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is \(^nC_r\).

Therefore, the number of subsets of A having r elements each is \(^nC_r\).

Hence, the total number of subsets of A

= \(^nC_0\) + \(^nC_1\) + \(^nC_2\) + …. + \(^nC_n\) = \((1 + 1)^n\) = \(2^n\).

[ Using binomial theorem ]

Leave a Comment

Your email address will not be published.