Additive functions, unions, and intersections.

[reb659]

Homework Statement

A function G:P--->R where R is the set of real numbers is additive provided
G(X1 U X2)=G(X1)+G(X2) if X1, X2 are disjoint.

Let S be a set, Let P be the power set of S. Suppose G is an additive function mapping P to R. Prove that if X1 and X2 are ARBITRARY(not necessarily disjoint subsets of W), then
G(X1 U X2)=G(X1)+G(X2)-G(X1 I X2)

Homework Equations

The Attempt at a Solution

The only way I know how to do this is using an element chasing proof. But if I let an element c be in the right hand side I can't go anywhere because the sets are not necessarily disjoint.

 

[arkajad]

Try using
$$X1\cup X2=X1\setminus X2+X2\setminus X1+X1\cap X2$$
$$X1=X1\setminus X2+X1\cap X2$$
and similar for X2. Draw it, play with the equations using additivity of f. I used "+" for disjoint unions.

 

[reb659]

I am such an idiot. I tried it before representing the left hand side as a union of disjoint sets but for some reason I didn't bother manipulating the right hand side in the same way.

Thanks a ton!