Proving Properties of a Nested Family of Sets

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SUMMARY

The discussion focuses on proving properties of a nested family of sets, specifically addressing two key statements: the equality of the union of sets and the intersection of sets. The first proof demonstrates that the union from k=1 to infinity of A(k) equals A(l), while the second proof shows that the intersection from k=1 to n of A(k) equals A(n). Essential to these proofs is the concept of subsets, as participants emphasize the necessity of demonstrating mutual subset relationships to establish equality.

PREREQUISITES
  • Understanding of nested families of sets
  • Familiarity with set operations: union and intersection
  • Knowledge of subset definitions and properties
  • Basic proof techniques in set theory
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  • Study the properties of nested families of sets in detail
  • Learn about the formal definitions of union and intersection in set theory
  • Explore proof techniques for establishing set equality
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Mathematicians, students studying set theory, and anyone interested in understanding advanced properties of sets and their proofs.

Willy_Will
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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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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?
 

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