I don't understand the question.

In summary, the conversation discussed a question regarding a mathematical expression involving sets and the use of De-Morgan rules. The speaker initially struggled to understand the equation, but eventually realized that it followed from the fact that one of the sets, S, was disjoint to the other sets A and B. The conversation ended with the speaker making a lighthearted comment about their ability to do math.
  • #1
Alone
60
0
This is a simple question.

On pages 5-6 of Measure Theory,Vol 1, Vladimir Bogachev he writes that:

for [tex]E=(A\cap S)\cup (B\cap (X-S))[/tex]

Now, he writes that:

[tex]X-E = ((X-A)\cap S) \cup ((X-B)\cap (X-S))[/tex]

But I don't get this expression, I get another term of [tex]((X-B)\cap (X-A))[/tex]

i.e, [tex]X-E =( ((X-A)\cap S) \cup ((X-B)\cap (X-S)))\cup ((X-B)\cap (X-A))[/tex].

I believe I did it correctly according to De-Morgan rules and distribution.

I am puzzled...:confused:
 
Last edited:
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  • #2
Re: Something in Measure Theory.

Nevermind, I got it.

It follows from the fact that S is disjoint to A and B.

Sometimes I wonder how I still can do math...:-D
 

FAQ: I don't understand the question.

1. What is Measure Theory?

Measure theory is a branch of mathematics that deals with the concept of measuring the size or extent of sets. It provides a rigorous framework for defining and measuring the size of both finite and infinite sets.

2. What are the fundamental concepts in Measure Theory?

The fundamental concepts in Measure Theory include measures, measurable sets, and integration. Measures are functions that assign a numerical value to sets, while measurable sets are sets that can be assigned a measure. Integration involves finding the area under a curve using measures.

3. What is the importance of Measure Theory in mathematics?

Measure Theory is an essential tool in many areas of mathematics, including probability, analysis, and geometry. It provides a solid foundation for understanding and working with abstract concepts, such as convergence, continuity, and integrals.

4. What are the different types of measures in Measure Theory?

There are several types of measures in Measure Theory, including Lebesgue measure, Borel measure, and Radon measure. Lebesgue measure is the most commonly used measure, and it assigns a measure to subsets of n-dimensional Euclidean space. Borel measure is a more general measure that can be applied to a wider range of sets, while Radon measure is used to measure more complicated sets, such as fractals.

5. How is Measure Theory used in real-world applications?

Measure Theory has many practical applications, especially in physics, engineering, and statistics. For example, it is used in statistical mechanics to calculate the probability of different states of a system, and in signal processing to analyze and manipulate signals. It is also used in computer science to measure the complexity of algorithms and data structures.

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