Hi

Would some one explain to a complete newbie how/why the set of even integers less than 100 IS NOT well ordered?

Thanks

Pravin

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- Oct 29th 2012, 04:43 PMpravinEven integers less than 100-Not Well Ordered ?
Hi

Would some one explain to a complete newbie how/why the set of even integers less than 100 IS NOT well ordered?

Thanks

Pravin - Oct 29th 2012, 05:40 PMPlatoRe: Even integers less than 100-Not Well Ordered ?
Well technically your statement is false.

Any set can be well ordered. We just don't know what the*order*may be.

That said, I think that your question assumes the*order*is the**usual one**.

Meaning that any nonempty subset has a least term.

It $\displaystyle A=\{2n:n\in\mathbb{Z}\text{ and }x<50\}$ does $\displaystyle A$ have a least term? - Oct 29th 2012, 09:40 PMpravinRe: Even integers less than 100-Not Well Ordered ?
If A does not have a least term could you explain why the smallest no. in the set is not the least term?

- Oct 30th 2012, 03:58 AMPlatoRe: Even integers less than 100-Not Well Ordered ?
The point is $\displaystyle A=\{2n:n\in\mathbb{Z}\text{ and }x<50\}$ can not have a least number. If $\displaystyle n\in A$ then $\displaystyle n-2\in A$ and $\displaystyle n-2<n$. We pick a number in $\displaystyle A$ we can find another number in $\displaystyle A$ smaller.

- Oct 31st 2012, 01:20 PMpravinRe: Even integers less than 100-Not Well Ordered ?
2 is the smallest even integer in the set (even integer less than 100) and why cannot 2 be the least element?

Peavin - Oct 31st 2012, 01:55 PMPlatoRe: Even integers less than 100-Not Well Ordered ?
- Oct 31st 2012, 08:36 PMpravinRe: Even integers less than 100-Not Well Ordered ?
Plato

Thank you very much for your help.

Pravin - Nov 1st 2012, 10:04 AMpravinRe: Even integers less than 100-Not Well Ordered ?
Another question- The set of positive even integers less than 100 is a well ordered set as it also has a least element?

- Nov 1st 2012, 10:21 AMPlatoRe: Even integers less than 100-Not Well Ordered ?
- Nov 1st 2012, 12:23 PMpravinRe: Even integers less than 100-Not Well Ordered ?
What about the set ofeven integers between -150 to +150. This should also be well-ordered.

- Nov 1st 2012, 01:20 PMPlatoRe: Even integers less than 100-Not Well Ordered ?
Any non-empty subset of $\displaystyle \mathbb{Z}$ which is bounded below contains a smallest term. If that is what you mean by well ordering then yes to your question.

One again as I said the first time, theoretically any set can be well ordered, we just don't know what the order would be.

But I think you mean the usual order $\displaystyle \le$. Saying that*any non-empty*set contains a first term. - Nov 1st 2012, 03:29 PMemakarovRe: Even integers less than 100-Not Well Ordered ?
In order to be well-ordered, it's not enough for the whole set to have the least element: every nonempty subset must have its own least element. I can create a nonstandard order on Z by taking the standard order and declaring 0 less than any other element of Z. With this order, Z has the least element, but it's still not a well-ordering.

Every nonempty subset of Z with the standard order that is bounded from below is well-ordered. - Nov 1st 2012, 09:17 PMpravinRe: Even integers less than 100-Not Well Ordered ?
Thanks. Let me write what I have understood from your reply.

1. Set of Positive integers from 5 to 75 is well ordered.

2. Set of Integers from -25 to 75 is well ordered? Yes or No

Please respond as this will confirm that I have understood it correctly.

Thanks once again

Pravin - Nov 1st 2012, 11:23 PMemakarovRe: Even integers less than 100-Not Well Ordered ?