Counting subsets

Question

(a) Please count how many functions
f : D → {0, 1} can be defined if the domain D is a finite set with the cardinality |D| = n.
(b) Can you find a bijection between the set of all such functions and the powerset P(D)?

Answer 1

The answer to part (a) is 2^n. A bijection between the functions and a powerset of D is the following: given a function f biject it with the element of the powerset of D that contains only elements d∈D for such f(d)=1. Here is another bijection: well order both sets and map the k-th element of the first set into the k-th element of the second set. There are (2^n)!-2 more bijections out there ;).