# Thread: Even integers less than 100-Not Well Ordered ?

1. ## Even 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

2. ## Re: Even integers less than 100-Not Well Ordered ?

Originally Posted by pravin
Would some one explain to a complete newbie how/why the set of even integers less than 100 IS 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 $A=\{2n:n\in\mathbb{Z}\text{ and }x<50\}$ does $A$ have a least term?

3. ## Re: 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?

4. ## Re: Even integers less than 100-Not Well Ordered ?

Originally Posted by pravin
If A does not have a least term could you explain why the smallest no. in the set is not the least term?
The point is $A=\{2n:n\in\mathbb{Z}\text{ and }x<50\}$ can not have a least number. If $n\in A$ then $n-2\in A$ and $n-2. We pick a number in $A$ we can find another number in $A$ smaller.

5. ## Re: 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

6. ## Re: Even integers less than 100-Not Well Ordered ?

Originally Posted by pravin
2 is the smallest even integer in the set (even integer less than 100) and why cannot 2 be the least element?
$-2$ is an even integer. $-2<2$ so $2$ is not the smallest even integer.

7. ## Re: Even integers less than 100-Not Well Ordered ?

Plato

Thank you very much for your help.

Pravin

8. ## Re: 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?

9. ## Re: Even integers less than 100-Not Well Ordered ?

Originally Posted by pravin
Another question- The set of positive even integers less than 100 is a well ordered set as it also has a least element?
Any nonempty subset of $\mathbb{Z}^+$ is well-ordered.

10. ## Re: Even integers less than 100-Not Well Ordered ?

What about the set ofeven integers between -150 to +150. This should also be well-ordered.

11. ## Re: Even integers less than 100-Not Well Ordered ?

Originally Posted by pravin
What about the set ofeven integers between -150 to +150. This should also be well-ordered.
Any non-empty subset of $\mathbb{Z}$ 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 $\le$. Saying that any non-empty set contains a first term.

12. ## Re: Even integers less than 100-Not Well Ordered ?

Originally Posted by pravin
Another question- The set of positive even integers less than 100 is a well ordered set as it also has a least element?
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.

13. ## Re: 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

14. ## Re: Even integers less than 100-Not Well Ordered ?

Originally Posted by pravin
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
Both sets are well-ordered.

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### set of even intgers less than 100 least element

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