Question

Let f(n) be the number of ways to take an n-element set s, and, if s has more than one element, to partition s into two disjoint nonempty subsets s1 and s2, then to take one of the sets s1, s2 with more than one element and partition it into two disjoint nonempty subsets s3 and s4, then to take one of the sets with more than one element not yet partitioned and partition it into two disjoint nonempty subsets, etc., always taking a set with more than one element that is not yet partitioned and partitioning it into two nonempty disjoint subsets, until only one-element subsets remain.

Find a simple formula for f(n). For instance, f(1) equals:

a) 0
b) 1
c) 2
d) 3

Answer 1

The function f(n) represents the number of ways to partition an n-element set into disjoint nonempty subsets until only one-element subsets remain, with a simple formula of f(n) = 2^(n-1).

Step-by-step explanation:

The function f(n) represents the number of ways to partition an n-element set into disjoint nonempty subsets until only one-element subsets remain. To find a simple formula for f(n), we can observe a pattern:

f(1) = 1

f(2) = 2

f(3) = 4

f(4) = 8

From this pattern, we can see that f(n) = 2^(n-1). Therefore, the answer is (b) 1.