# A well-ordered set

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

Question: Since $A$ is well ordered and $B$ 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 $B$ is not necessarily well-ordered?
• Mar 9th 2010, 11:31 AM
Plato
Quote:

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

Question: Since $A$ is well ordered and $B$ 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 $B$ is not necessarily well-ordered?

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

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

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

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

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.

Since the set $A$ of real numbers is a well-ordered set, by definition, every subset of $A$ must have a least element, and since $B$ is a subset of A, it follows that every subset of $B$ is a subset of $A$ containing a least element. In turn, every subset of $B$ has a least element. Consequently, by the same definition, $B$ is a well-ordered set.