(related to Proposition: Number of Subsets of a Finite Set)
Consider the set \(C=\{char(Y), Y\subseteq X\}\) of all characteristic strings defined on the set \(S:\{Y\subseteq X\}\). According to the proof of characteristic strings, we have \(|C|=|S|\). Note that each \(char(Y)\) is a string of the length \(|X|=n\) over the alphabet \(\Sigma=\{0,1\}\), (i.e. of \(2\) letters).
According to the rule about the number of strings with a fixed length \(n\) over an alphabet with \(k\) letters the number of such strings is \(k^n\). Therefore we have\[|S|=|C|=2^n.\]