Interchange Between Union and Intersection: Am I Correct?

In summary, the conversation is about the interchange between union and intersection in set operations and the possibility of using a change of variable technique. The participant is asking for clarification and additional resources on this topic.
  • #1
wayneckm
68
0
Hello all,


I have the following question regarding the interchange between union and intersection.

[tex] \cup_{q < t} \cap_{s > q} A_{s} = \cap_{s<t} \cup_{q<s} A_{q} = \cup_{q < t} A_{q}[/tex]

Am I correct? Also, can anyone provide me some more resources regarding this kind of interchange in set operation or change of variable technique?

Thanks.


Wayne
 
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  • #2
Off the top of my head, it doesn't look correct.

Suppose Ai is empty for all i>100. Then [tex]\cap_{s>q}A_s[/tex] will be empty for all q, and the union of these will be empty. So lhs is empty.

But rhs, being just a union of As, will not necessarily be empty.
 

1. What is the difference between union and intersection?

The union of two sets A and B is a set that contains all elements that are in A or in B (or both). The intersection of two sets A and B is a set that contains all elements that are in both A and B.

2. Can the union and intersection of two sets be the same?

Yes, the union and intersection can be the same if the two sets are identical.

3. How do you represent union and intersection in mathematical notation?

The union of sets A and B is represented as A ∪ B, while the intersection is represented as A ∩ B.

4. What is the significance of interchange between union and intersection?

The interchange between union and intersection is significant because it allows for the distribution of operations over sets, making it easier to solve complex problems involving sets.

5. Are there any real-life applications of interchange between union and intersection?

Yes, interchange between union and intersection is commonly used in statistics, probability, and computer science to solve problems involving sets and their operations.

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