A question on theme of well-ordered sets.
Lets say I have a set that contains the all odd natural numbers.
Also, a set .
Given that .
Let be a first element in in .
We define the order on be the set:
I need to show that is first element in in
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I started by saying that if , and odd, then , hence is first.
But what I do with the case in which is even...?
Thank you for your time.
I did a mistake on the original post of mine, not surprising fact...
Lets say I have a set that contains the all odd natural numbers.
Also, a set .
Given that .
Let be a first element in in .
We define the order on be the set:
I need to show that is first element in in
-------------------------------------------------------------------------------
I started by saying that if , and odd, then , hence is first.
But what I do with the case in which is even...?
Thanks again!
Answers:
1. The first element in is 0.
2. in is the set: . And 0 is first.
3. I'm trying to prove that the ordered set is well ordered.
So what I have done so far:
Let be a set of all even numbers. , hence well ordered.
Let be a set of all odd numbers. , hence well ordered.
Let be non-empty subset of .
Let we look on . Suppose . Let be first element in , in well ordered set .
is first element in in .
Explanation for my last sentence:
For all element of which is different from , if is even, then , hence is first.
If is odd, is first by definition of order .
Now, if then .
Let be first element in in .
How can I show that is first element in in ?
Thanks again!
This is still very confusing to me. Are you saying that is just the ordering which can be explained 'First and then "? And, so what you're trying to show is that if one considers as a subset of that it has a least element...and moreover that its least element coincides with its least element as a subset of with the usual ordering?
Let be the subset of even numbers in and
Now the order in works this way: any two even or odd numbers are ordered in the natural way.
But any even number precedes every odd number.
Now consider any nonempty subset in .
If then the first even in is its first element.
Else wise, the first odd number in is its first element.