Finding the Measure of A, B: Prove m(A)+m(B)=m(AuB)+m(AnB)

  • Thread starter pivoxa15
  • Start date
  • Tags
    Measure
In summary, the conversation discusses proving a statement involving a sigma algebra and the measure of sets. The main strategy is to use the property that a measure is additive on countable disjoint unions, and to write the sets in the statement as unions of disjoint sets. The conversation also includes some hints on how to approach the problem.
  • #1
pivoxa15
2,255
1

Homework Statement



A, B in a sigma algebra

Prove
m(A)+m(B)=m(AuB)+m(AnB)

m denotes the measure.

The Attempt at a Solution



Don't see how to do it.

Somehow we are dealing with each individual set and taking the measure on them. Then finding what they equate to.
 
Physics news on Phys.org
  • #2
A measure is additive on countable disjoint unions. So have you tried to write A[itex]\cup[/itex]B as a union of disjoint sets?

(Note: I'm assuming m is a finite measure, i.e. does not take on infinity.)
 
Last edited:
  • #3
Ok you haven't tried hard Pivoxa
Hint:
use [tex] A= [A - (A \cap B)] \cup (A \cap B) \hspace{10pt}\text{or}\hspace{10pt} B= [B - (A \cap B)] \cup (A \cap B)[/tex] and see what you get from it.
 

1. What does "m" stand for in the equation?

The "m" stands for the measure of the given sets A, B, A∪B (union of A and B), and A∩B (intersection of A and B).

2. How is the measure of A, B related to the measure of their union and intersection?

The equation m(A)+m(B)=m(A∪B)+m(A∩B) is known as the additivity property of measures. This means that the measure of the union of two sets is equal to the sum of their individual measures minus the measure of their intersection.

3. Is this equation always true for any sets A and B?

Yes, this equation is true for any sets A and B. It is a fundamental property of measures that holds for both finite and infinite sets.

4. Can this equation be used to find the measure of a set if the measures of its subsets are known?

Yes, this equation can be rearranged to find the measure of a set A or B if the measures of their union and intersection are known. For example, if m(A), m(B), and m(A∪B) are known, the measure of their intersection m(A∩B) can be found by rearranging the equation as m(A∩B) = m(A)+m(B)-m(A∪B).

5. How is this equation used in real-life applications?

This equation is used in various fields of science and mathematics, such as geometry, probability, and statistics. It is also commonly used in real-life situations to calculate the probability of events, determine the area of overlapped regions, and find the total amount of a substance in a mixture.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
547
  • Calculus and Beyond Homework Help
Replies
16
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
953
  • Calculus and Beyond Homework Help
Replies
1
Views
4K
  • Calculus and Beyond Homework Help
Replies
3
Views
718
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
451
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
828
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
Back
Top