A Noncombinatorial Proof of the Number of All Subsets of a Finite Set
Manmohan S. Arora, Jerry D. Allen · Mathematics Teacher Learning and Teaching PK-12 · 1970
Given a finite set Sn = {a1, a2, …, an}, we denote the set of all subsets of Sn by 2Sn, and prove by induction that #(2Sn) = 2n, where “#” denotes “the cardinality of.” If n = 1, S1 = {a1}, and clearly 2S1 = {ø, S1}.