MathDB

Define set

S_n=\{1,2,\cdots,n\}

X

is a subset of

S_n

. We call sum of all numbers in

X

capacity (capacity of empty set is

0

). If capacity of

X

is odd/even, then we call it odd/even subset. (a) Prove that the number of odd subsets and even subsets of

S_n

are the same. (b) Prove that the sum of capacity of all odd subsets and even subsets are the same when

n\geq3

. (c) Calculate the sum of capacity of all odd subsets when

n\geq3

Problem Statement