# Question Why does the power set of a finite set with n elements have exactly $2^n$ elements? What is the underlying principle that explains this? ## Answer Choices (A) Because each element can either be included or excluded from a subset, giving 2 choices per element (B) Because the power set includes all possible unions of elements (C) Because exponential growth is a property of all set operations (D) Because the power set formula is defined to be $2^n$ by convention (E) Because pairs of subsets can always be formed from n elements # Solution The power set $\mathcal{P}(A)$ of a set A is the set of all subsets of A, including ∅ and A itself. For a set with n elements, we can think of constructing each subset by making a binary decision for each element: include it or don't include it. Since there are n elements and 2 choices per element, by the multiplication principle, there are $2 \times 2 \times \cdots \times 2$ (n times) = $2^n$ total possible subsets. More formally, this can be proven using bijection. Each subset corresponds to a unique binary string of length n (where the i-th position is 1 if the i-th element is included, 0 otherwise). There are exactly $2^n$ such binary strings, establishing a one-to-one correspondence between subsets and binary strings. This is not a definition or convention—it's a theorem that follows from the combinatorial structure of subset formation. The other options either mischaracterize the reason (unions, pairs) or incorrectly suggest it's definitional rather than derivable. **Answer: (A)**