Family of sets without maximal element

In summary, the individual is asking about maximal elements from a linear algebraic perspective and has a few questions about families and chains of sets having a maximal element. However, it is not necessary for every family or chain to have a maximal element, as shown by the counter-example of A_n= [-n, n]. This highlights the importance of Zorn's lemma in determining the existence of maximal elements.
  • #1
Bipolarity
776
2
I have begun to learn about maximal elements from a linear algebraic perspective (maximal linearly independent subsets of vector spaces). I have a few questions of which I have been able to get few insights online:

1) Does every family of sets have a maximal element? How can I make a family of sets that does not have a maximal element? I have to obviously make the hypothesis of Zorn's lemma fail, but I can't quite see how to do that.

2) Does every chain of sets have a maximal element? It seems that a chain of sets necessarily satisfies the criteria for Zorn's lemma but I am not sure.

Thanks!

BiP
 
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  • #2
No, to both of those. A counter-example is [tex]A_n= [-n, n][/tex], the closed interval from -n to n.

Zorns lemma says that if every chain in an ordered collection has a maximum, then the collection has maximal elements. If it were true that every chain has a maximum or that every family has a maximum, Zorn's lemma would be unnecessary.
 

1. What is a family of sets without maximal element?

A family of sets without maximal element is a collection of sets where there is no single set that includes all the other sets in the family. This means that there is no set in the family that is the largest or contains all the other sets as subsets.

2. How is a family of sets without maximal element different from a family of sets with a maximal element?

A family of sets without maximal element is different from a family of sets with a maximal element because in the latter, there is a largest set that contains all the other sets in the family. In a family without a maximal element, there is no such set.

3. Can a family of sets without maximal element have an infinite number of sets?

Yes, a family of sets without maximal element can have an infinite number of sets. As long as there is no single set that contains all the other sets, the family can have any number of sets.

4. What is an example of a family of sets without maximal element?

An example of a family of sets without maximal element is the collection of all open intervals on the real number line. Each interval is a set, but there is no single interval that contains all the other intervals in the family. Therefore, this is a family of sets without a maximal element.

5. Why is the concept of a family of sets without maximal element important in mathematics?

The concept of a family of sets without maximal element is important in mathematics because it allows for the study of infinite collections of sets that do not have a largest or all-encompassing set. This is useful in various areas of mathematics, such as topology and measure theory, where infinite collections of sets are commonly studied.

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