set and subset question Given  a nonempty set S, we select a fixed subset $S_0\subseteq S$ and use it to define two functions on the set of all subsets of $S$:


$f(A)=A\cup S_0$
$g(A)=A\setminus S_0$


For what choices of $S_0$ do these functions commute? 


Only for $S_0=\varnothing$? or
Only for $S_0=S$? or
For all $S_0$?
HINT:

Recall that the functions commute if for every $A$, $f(g(A))=g(f(A))$, that is $$(A\cup S_0)\setminus S_0=(A\setminus S_0)\cup S_0.$$

The choices for which the functions don't commute can be found by testing the case where $S=\{a,b,c\}$. The choice where the functions do commute is easily provable.
$gf(A)=g(A\cup S_0)=(A\cup S_0)\setminus S_0=A\setminus S_0$

$fg(A)=(A\setminus S_0)\cup S_0=A$

1 -true