0 votes 0 votes If C is a set with c elements, how many elements are in the power set of C? Explain your answer. Theory of Computation michael-sipser theory-of-computation set-theory easy + – admin asked Apr 13, 2019 admin 2.7k views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes The power set will have $2^c$ elements. This is because while generating the powet set, we have two choices for each element in the original set. Hence, $\underbrace{2 \times 2 \times \dots 2}_{\text{c times }} = 2^c$. goxul answered Apr 14, 2019 goxul comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes One well know way is already there in answer. Here is another way. The number of subsets with k elements in the power set of a set with n elements is given by the number of combinations, C(n, k), also called binomial coefficients. For example, the power set of a set with three elements, has: C(3, 0) = 1 subset with 0 elements (the empty subset), C(3, 1) = 3 subsets with 1 element (the singleton subsets), C(3, 2) = 3 subsets with 2 elements (the complements of the singleton subsets), C(3, 3) = 1 subset with 3 elements (the original set itself). Using this relationship we can compute using the formula: Therefore, one can deduce the following identity, assuming ${\textstyle |S|=n}$ Source: https://en.wikipedia.org/wiki/Power_set#Relation_to_binomial_theorem smsubham answered Dec 26, 2019 smsubham comment Share Follow See all 0 reply Please log in or register to add a comment.