For a fixed set $B$ and for sets $A_i ,\forall i \in \left \{ 1,2,\dots,n \right \}$ , I define $f(A_i)=\frac{|A_i \cap B|}{2|A_i|-|A_i\cap B|}$, where $|A_i|>0$ is the cardinality of set $A_i$.
Is $f(A_i)$ submodular? i.e, is $f(A_i)+f(A_j)\geq f(A_i\cup A_j)+f(A_i\cap A_j), \quad \forall i,j \in \left \{ 1,2,\dots,n \right \}$
Is $f(A_i)$ submodular in a sub-case when $A_i \cap A_j=\varnothing,\forall i,j$ , $f(\varnothing)=0$ and $|A_i\cap B|>0,\forall A_i$ ? If the above is proved directly, the sub-case also follows.