# Thread: A well-ordered set

1. ## 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?

2. 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 $\displaystyle A?$

3. 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?

4. 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.

Here is the answer:

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

I think you can be proud of me.

5. Remember it is every nonempty subset...

6. Thank you, Plato, I will remember not to leave out the word "non-empty,"

Oh, I would have lost my case at court.