MathDB

Let

S

be a finite set.

f

is a function defined on the subset-group

2^S

of set

S

f

is called \textsl{monotonic decreasing} if when

X \subseteq Y\subseteq S

, then

f(X) \geq f(Y)

holds. Prove that:

f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)

for

X, Y \subseteq S

if and only if

g(X)=f(X \cup \{ a \}) - f(X)

is a \textsl{monotonic decreasing} funnction on the subset-group

2^{S \setminus \{a\}}

of set

S \setminus \{a\}

for any

a \in S

Problem Statement