# Even integers less than 100-Not Well Ordered ?

• October 29th 2012, 05:43 PM
pravin
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
• October 29th 2012, 06:40 PM
Plato
Re: Even integers less than 100-Not Well Ordered ?
Quote:

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?
• October 29th 2012, 10:40 PM
pravin
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?
• October 30th 2012, 04:58 AM
Plato
Re: Even integers less than 100-Not Well Ordered ?
Quote:

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.
• October 31st 2012, 02:20 PM
pravin
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
• October 31st 2012, 02:55 PM
Plato
Re: Even integers less than 100-Not Well Ordered ?
Quote:

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.
• October 31st 2012, 09:36 PM
pravin
Re: Even integers less than 100-Not Well Ordered ?
Plato

Thank you very much for your help.

Pravin
• November 1st 2012, 11:04 AM
pravin
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?
• November 1st 2012, 11:21 AM
Plato
Re: Even integers less than 100-Not Well Ordered ?
Quote:

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.
• November 1st 2012, 01:23 PM
pravin
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.
• November 1st 2012, 02:20 PM
Plato
Re: Even integers less than 100-Not Well Ordered ?
Quote:

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.
• November 1st 2012, 04:29 PM
emakarov
Re: Even integers less than 100-Not Well Ordered ?
Quote:

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.
• November 1st 2012, 10:17 PM
pravin
Re: Even integers less than 100-Not Well Ordered ?
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
• November 2nd 2012, 12:23 AM
emakarov
Re: Even integers less than 100-Not Well Ordered ?
Quote:

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.