Proving Properties of a Nested Family of Sets

In summary, the conversation is discussing how to prove that two sets, U (from k=1 to infinity) A(sub k) and A(sub l), are equal. The participants suggest using the fact that they are subsets of each other and the assumption of nested sets to prove their equality. They also mention using the concepts of unions and intersections of sets to further solidify the proof.
  • #1
Willy_Will
15
0
Hi all,

Im having trouble with this problem, I don't know where to begin.

Suppose that Ä is a nested family of sets.

1. Prove that U (from k=1 to infite) A(sub k) = A(sub l)

2. Prove that ∩ (from k=1 to n) A(sub k) = A(sub n)


Thanks in advace mathematicians!

-William
 
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  • #2
1) To show that they are equal you must show that they are subsets of one another.

2) If x is in the union of a collection of sets, what does this mean?

3) If x is in the intersection of a collection of sets, what does this mean?

4) You have to use the assumption of nested somewhere.

5) If A is a subset of B what can you say about A n b?

6) If A is a subset of B what can you say about A U B?
 
  • #3


Hi William,

To prove these properties, we need to use the definition of a nested family of sets and the properties of unions and intersections. Let's start with the first property:

1. Prove that U (from k=1 to infite) A(sub k) = A(sub l)

To prove this, we need to show that every element in the union on the left-hand side is also in the set on the right-hand side and vice versa. By definition, the union of a nested family of sets Ä is the set of all elements that are in at least one of the sets in the family. So, if an element is in the union, it must be in at least one of the sets A(sub k). But since the sets are nested, this means that the element is also in the set A(sub l), since A(sub l) contains all the elements in A(sub k). Similarly, if an element is in A(sub l), it is also in every other set in the nested family, and therefore in the union. This shows that the two sets are equal.

2. Prove that ∩ (from k=1 to n) A(sub k) = A(sub n)

Similarly, to prove this property, we need to show that every element in the intersection on the left-hand side is also in the set on the right-hand side and vice versa. By definition, the intersection of a nested family of sets is the set of all elements that are in every set in the family. So, if an element is in the intersection, it must be in every set A(sub k). But since the sets are nested, this means that the element is also in the set A(sub n), since A(sub n) contains all the elements in A(sub k). Similarly, if an element is in A(sub n), it is also in every other set in the nested family, and therefore in the intersection. This shows that the two sets are equal.

I hope this helps to clarify the properties of nested families of sets. Remember to always use the definitions and properties of unions and intersections to prove these types of properties. Good luck with your studies!
 

1. What is a nested family of sets?

A nested family of sets refers to a collection of sets where each set contains all the elements of the previous set, and each set is a subset of the next set. This creates a hierarchical structure, with the largest set containing all the elements and the smallest set containing the fewest elements.

2. Why is it important to prove properties of a nested family of sets?

Proving properties of a nested family of sets is important because it helps to establish the relationships between the sets and their elements. It also allows us to make logical deductions about the sets and their elements, which can be useful in various fields such as mathematics, computer science, and data analysis.

3. What are some common properties of a nested family of sets?

Some common properties of a nested family of sets include the nested set property, which states that the intersection of all the sets in the family is equal to the smallest set in the family. Another common property is the inclusion-exclusion property, which states that the cardinality of the union of all the sets in the family is equal to the sum of the cardinalities of each set minus the cardinality of their intersection.

4. How are properties of a nested family of sets proven?

Properties of a nested family of sets can be proven using various methods such as mathematical induction, proof by contradiction, or by using set identities and properties. It is important to carefully analyze the relationships between the sets and their elements and use logical reasoning to construct a proof.

5. What are some real-world applications of nested family of sets?

Nested families of sets have many applications in different fields. In mathematics, they can be used to prove theorems and solve problems in set theory and topology. In computer science, they are used in data structures such as trees and graphs. In data analysis, they can be used to organize and classify data. They also have applications in various engineering and scientific fields, such as control theory and systems analysis.

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