Independence between random variables

Question

Say I have four random variables: $X$, $Y$, $A$, and $B$. Now say that $X$ and $A$ are independent, $X$ and $B$ are independent, $Y$ and $A$ are independent, and $Y$ and $B$ are independent. Does this imply that $f(X, Y)$ and $g(A, B)$ are independent for some real functions $f$ and $g$? Note that $X$ and $Y$ may not be independent, same with $A$ and $B$.

Answer 1

Independence cannot be handled by intuition alone. The answer to the question is NO! For example we can have events $D,E,F$ such that any two of them are independent but the three together are not. To simplify things take $Y=0$. It is given that X is independent of A as well as B. For X to be independent of $g(A,B)$ for any measurable function g it is necessary that X is jointly independent of X and Y and this may not be true. In particular take the three events $D,E,F$ mentioned above, take $X=I_D, A=I_E, B=I_F$ Let $g(a,b)=ab$. Then X and $g(A,B)$ are not independent.

Answer 2

I can think of the following counter example. Let $$X=U$$$$Y=U\oplus W$$$$A=V$$$$B=V\oplus W$$ where $U,V,W$ are i.i.d. uniform in $\{0,1\}$ and $\oplus$ stands for modulo-$2$ addition. It is easy to see that all the required independence relations are satisfied.

Now assume $$f(x,y)=g(x,y)=x\oplus y.$$ Then we get $$f(X,Y)=g(A,B)=W$$ and hence, $f(X,Y)$ and $g(A,B)$ are not independent.