# A well-ordered set

• Mar 9th 2010, 11:15 AM
novice
A well-ordered set
If \$\displaystyle A\$ is any well-ordered set of real numbers, then every subset of \$\displaystyle A\$ has a least element. If \$\displaystyle B\$ is a nonempty subset of \$\displaystyle A\$, then \$\displaystyle B\$ has a least element.

Question: Since \$\displaystyle A\$ is well ordered and \$\displaystyle B \$ has a least element, and since having a least element is not a sufficient condition for a nonempty set to be well-ordered, does it mean that \$\displaystyle B\$ is not necessarily well-ordered?
• Mar 9th 2010, 11:31 AM
Plato
Quote:

Originally Posted by novice
If \$\displaystyle A\$ is any well-ordered set of real numbers, then every subset of \$\displaystyle A\$ has a least element. If \$\displaystyle B\$ is a nonempty subset of \$\displaystyle A\$, then \$\displaystyle B\$ has a least element.

Question: Since \$\displaystyle A\$ is well ordered and \$\displaystyle B \$ has a least element, and since having a least element is not a sufficient condition for a nonempty set to be well-ordered, does it mean that \$\displaystyle B\$ is not necessarily well-ordered?

If \$\displaystyle B\subseteq A\$, is it true that each subset of \$\displaystyle B\$ is a subset of \$\displaystyle A?\$
• Mar 9th 2010, 12:00 PM
novice
Quote:

Originally Posted by Plato
If \$\displaystyle B\subseteq A\$, is it true that each subset of \$\displaystyle B\$ is a subset of \$\displaystyle A?\$

Beside it being a subset of \$\displaystyle A\$, what is the reason to believe the set \$\displaystyle B\$ is well-ordered?
• Mar 9th 2010, 12:56 PM
novice
Quote:

Originally Posted by Plato
If \$\displaystyle B\subseteq A\$, is it true that each subset of \$\displaystyle B\$ is a subset of \$\displaystyle A?\$

Plato,

I honestly have spent the entire time, since your last reply, thinking of the question you put forth. I think I know what you were trying to show me, but I want to know if I understood it well.