2

consider following question

I am able to easily see that set of positive rationals is not well ordered set but I have difficulty coming to conclusion with this

set of positive rationals with denominator less than 200

Is this well ordered set or not ?

I think it is well ordered set .If it is not, could someone give me non empty subset that is not bounded from below

another question regarding well ordered sets I read from wiki article that "the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element."I got that

Another relation for well ordering the integers is the following definition: x ≤z y iff (|x| < |y| or (|x| = |y| and x ≤ y)). This well order can be visualized as follows:

0 −1 1 −2 2 −3 3 −4 4 ...

how can this be well ordered set, when we use argument same as above

consider set of negative integers -1 -2 -3 -4 -5 but this is not bounded from below, so how can this be well ordered set

manifold
  • 1,485
  • 1
    Under the ordering provided, the negative integers have a minimum of $-1$, as if $x$ is a negative integer, then $|-1| < |x|$ for $x \neq -1$, or $|x| = |-1|$ and $x \le -1$ for $x = -1$. – Theo Bendit Jul 10 '17 at 01:14

2 Answers2

3

Hint:

Reduce all these rational numbers to denominator $200!$.

Bernard
  • 175,478
  • so that would be 1/200 2/200 .....199/200,1,201/200,202/200..... so every subset is bounded from below. So, it is well ordered set. Correct me if I am wrong – manifold Jul 10 '17 at 00:50
  • 1
    @viru He talked about $200!$, not $200$. This is important as $1/199$ would not be in that list, but you can write it as $\frac{200 \cdot 198!}{200!}$. – Theo Bendit Jul 10 '17 at 01:28
-1

ThIs is the conclusion for the set of rational numbers less than or equal to 200 not been well order..

Let $X= \{u: u\text{ is a rational number and }0 \ll u\ll200 \}$ and

$E=\{1/n : n \in \Bbb N\setminus 0\}$ which is a subset of $X$.

Hence there exist no first element --> not well ordered.

Daniele Tampieri
  • 9,727
Folebe
  • 1
  • You seem to have missed the fact that the question is about the set of rational numbers which can be written as $\frac{a}{b}$ for some $a\in\Bbb Z$, $b\in\Bbb N$ and $0<b\le 200$. –  Feb 03 '19 at 19:57