Munkres’ Proof of Well Ordering Property Code Answer

Hello Developer, Hope you guys are doing great. Today at Tutorial Guruji Official website, we are sharing the answer of Munkres’ Proof of Well Ordering Property without wasting too much if your time.

The question is published on by Tutorial Guruji team.

In Munkres’ Topology the proof for the Well-Ordering Property is stated as follows:
enter image description here

  • I’m having confusion with the first and second underlined part: “Let A be the set of all positive integers n for which the statement holds.” Isn’t this the very statement that we are trying to show is true in this subproof ? Can you please restate this in a different way because I feel as though it might be ambigious??
  • When he says “this set” in the last underlined part does he mean the set $C cap {1,…, n }$ or the set $A$?
  • And I’m sorry to put it so generally but what is the key idea behind the subproof; is it to show that intersection between the set of the first $n$ integers and any subset of it has a least element? I don’t see how such an intersection should imply the existence of a least element.

Thanks in advance guys, I hate asking to clarify so many little points but this one just ain’t clicking for me.

Answer

I’m having confusion with the first and second underlined part: “Let A be the set of all positive integers n for which the statement holds.” Isn’t this the very statement that we are trying to show is true in this subproof ? Can you please restate this in a different way because I feel as though it might be ambigious??

No, this statement is merely a definition. The symbol $A$ is being defined to refer to the set of $n$ such that the statement holds. We don’t yet know that the statement holds for all $n$, and we’re just collecting all the $n$ such that it does hold into a set and calling that set $A$. We’re not asserting that the statement ever actually does hold; maybe $A$ is empty.

The role of $A$ is that we now want to prove $A=mathbb{Z}_+$. That means that every positive integer is an element of $A$, or in other words that the statement holds for every positive integer $n$.

When he says “this set” in the last underlined part does he mean the set $C cap {1,…, n }$ or the set $A$?

He means $C cap {1,…, n }$. The statement $nin A$ by definition means that any nonempty subset of ${1,dots,n}$ has a least element. Since $Ccap {1,dots,n}$ is a nonempty subset of ${1,dots,n}$, it therefore has a least element.

And I’m sorry to put it so generally but what is the key idea behind the subproof; is it to show that intersection between the set of the first $n$ integers and any subset of it has a least element? I don’t see how such an intersection should imply the existence of a least element.

I’m not quite sure what you’re asking here. The overall goal is to prove that if $D$ is any nonempty subset of $mathbb{Z}_+$, then it has a least element. We do this using the following observation: if $nin D$, then the least element of $D$ must be less than or equal to $n$. So if we find the least element $k$ of $Dcap {1,dots,n}$, then $k$ is actually the least element of all of $D$. Indeed, for any $din D$, if $dleq n$, then $kleq d$ since $din Dcap{1,dots,n}$ and $k$ is the least element of $Dcap{1,dots,n}$. And if $d>n$, then obviously $kleq d$ since $kleq n$.

So using this idea, we only need to prove that $Dcap {1,dots,n}$ has a least element. Now we rename $Dcap{1,dots,n}$ to $C$ and think of $C$ as just some arbitrary nonempty subset of ${1,dots,n}$. So what we want to prove now is that any nonempty subset of ${1,dots,n}$ has a least element. This is what the subproof is proving, and it does so by induction on $n$.

We are here to answer your question about Munkres’ Proof of Well Ordering Property - If you find the proper solution, please don't forgot to share this with your team members.

Related Posts

Tutorial Guruji