Exercise 1.7.1

Answers

1.: No. For example, the family ${(0, n)}_{n \geq 1}$ of open intervals has no maximal element.
2.: No. For example, the family ${(0, n)}_{n \geq 1}$ of open intervals in the set real numbers has no maximal element.
3.: No. For example, the two set in this family ${1, 2, 2, 3}$ are both maximal element.
4.: Yes. If there are two maximal elements $A$ and $B$ , we have $A \subset B$ or $B \subset A$ since they are in a chain. But no matter $A \subset B$ or $B \subset A$ implies $A = B$ since they are both maximal elements.
5.: Yes. If there are some independent set containing a basis, then the vector in that independent set but not in the basis cannot be a linear combination of the basis.
6.: Yes. It’s naturally independent. And if there are some vector can not be a linear combination of a maximal independent set. We can add it to the maximal independent set and have the new set independent. This is contradictory to the maximality.

jephianlin

2011-06-27 00:00