# Additive functions, unions, and intersections.

• reb659
In summary, the conversation discusses the proof of an additive function G mapping the power set P of a set S to the set of real numbers. It is proven that for arbitrary subsets X1 and X2 of S, G(X1 U X2) = G(X1) + G(X2) - G(X1 I X2). The discussion includes the use of element chasing and manipulation of equations using the additivity property of G.
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)

## 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.

Last edited:
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.

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!

## 1. What are additive functions?

Additive functions are mathematical functions that satisfy the property of additivity, meaning that the function's output for the sum of two inputs is equal to the sum of the outputs for each individual input. In other words, f(x+y) = f(x) + f(y).

## 2. How are unions and intersections different?

Unions and intersections are both set operations in mathematics. A union combines two or more sets to create a new set that contains all the elements from the original sets. An intersection, on the other hand, only includes elements that are common to all the sets being intersected.

## 3. Can a function be both additive and non-linear?

No, a function cannot be both additive and non-linear. Additive functions must satisfy the property of additivity, while non-linear functions do not have a constant rate of change and do not follow a straight line on a graph. Therefore, a function cannot have both of these properties at the same time.

## 4. How do you determine the union or intersection of more than two sets?

To determine the union of more than two sets, you can simply combine all the elements from each set into one new set. To determine the intersection of more than two sets, you can find the common elements among all the sets being intersected.

## 5. How are additive functions used in real life?

Additive functions are used in many real-life applications, such as in economics, physics, and engineering. In economics, additive functions are used to model consumer demand and supply. In physics, they are used to describe the relationship between force and displacement. In engineering, they are used to calculate the total energy in a system.

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