All Questions
Tagged with well-orders solution-verification
37
questions
0
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88
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Iterative argument in math proofs
I am reading proof of subspace of $\mathbb{R^n}$ has a basis. And most of them like this(classic proof): let $S\subset\mathbb{R^n}$ ,if $v_1\neq0$ and $<v_1>=S$, we finished proof, otherwise, we ...
- 117
2
votes
1
answer
42
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If $X_1$,$X_2$ are wosets isomorphic to ordinals $\alpha_1,\alpha_2$ then $X_1\times X_2$ is isomorphic to $\alpha_2\cdot \alpha_1$
I want to prove the following:
Let $X_1$,$X_2$ be wosets, isomorphic to ordinals $\alpha_1,\alpha_2$ respectively. Then $X_1\times X_2$, with the lexicographic order, is isomorphic to $\alpha_2\cdot\...
- 4,827
1
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a proof idea: Every well-ordered set has an order-preserving bijection to exactly one ordinal.
I have seen a proof of the statement, and its usually by transfinite induction. And I'm trying to find out why my proof doesn't work, it seems too simple:
Let $X$ be a well-ordered set. Define $X^{<...
- 438
6
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1
answer
264
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Does there exist an infinite set admitting precisely four linear orders?
I am studying a course on ZF Set Theory where I encountered the following question:
Does there exist a set admitting precisely four linear orders?
Since any set with $n$ elements exhibits $n!$ ...
- 4,331
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1
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59
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Well Ordering Principle Proof without mathematical induction with different approach
Denote $\Bbb Z_0$ be the set of all non-negative integers.
Well Ordering Principle for $\Bbb Z_0$. Every non-empty subset $S$ of $\Bbb Z_0$ has a least element; that is, there exists $m \in S$ such ...
- 447
3
votes
2
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324
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Product of Countable Well-Ordered Set with $[0,1)$ is Homeomorphic to $[0,1)$
As part of a proof that the long line is locally Euclidean, I'd like to prove the following:
Proposition. If $A$ is a countable well-ordered set, then $A \times [0,1)$ with the dictionary order is ...
- 2,221
1
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1
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75
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How do you prove that there exists a highest element of any finite, nonempty subset of Natural Numbers? Is the following algorithmic proof valid?
Since the given set, $C \subset \mathbb{N}$ is non empty, hence by well ordering principle there exists $\alpha \in C$ which is the lowest element in C. Also, since the set $C$ is finite, $\quad \...
- 308
1
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108
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Discussion of Exercise 9, section 4 on page 35 of Munkres’ Topology 2E.
In Exercise 9, section 4 on page 35 of Munkres’ Topology 2E, the problem is stated as follows.
Exercise 4.
(a) Show that every nonempty subset of $\mathbb{Z}$ that is bounded above has a largest ...
- 307
1
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46
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Discussing Ex 13. Section 3. p-29, Munkres' Topology 2E. [duplicate]
In Exercise 13, section 3 on page 29 of Munkres’ Topology 2E, the problem is stated as follows.
Prove the following:
Theorem. If an order set $A$ has the least upper bound property, then it has the ...
- 307
1
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0
answers
113
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Does any non-maximal element of a well-ordered set have a unique successor?
$\newcommand{\setcomplement}[2]{#1 \setminus #2}$
$\newcommand{\singleton}[1]{\left\{#1\right\}}$
$\newcommand{\segment}[2]{\operatorname{Seg}_{#1}\left(#2\right)}$
I wanted to find the smallest ...
- 1,840
1
vote
1
answer
95
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Why is this step required in the proof of sum of first $n$ odd numbers using the Well Ordering Principle?
I came across this question while doing $\text{6.042J}$ from MITOCW. I have a doubt in the part c, namely, why do we need to manipulate the formula in that way?
Here is my solution so far to the ...
- 1,454
2
votes
1
answer
83
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Showing that 49¢ is not makeable using the given conditions
While going through 6.042J from MITOCW, in the text Mathematics for Computer Science, I came across the following problem at which I'm stuck.
Now, I proceeded doing the proof in the following manner.
...
- 1,454
1
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0
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24
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Prove that if W is an initial segment of X×Y then there exists an initial segment V in Y such that X×V is an initial segment of X×Y containing W
Given a ordered set $(X,\preceq)$ for any $\xi\in X$ we call the set
$$
I_\xi:=\{x\in X:x⪱\xi\}
$$
the initial segment of $\xi$.
Now if $(X,\preceq)$ and $(Y,\precsim)$ are two ordered sets then it is ...
- 4,906
1
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0
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85
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State if each of the objects' set is well-defined or not.
Q. 1.1 taken from book titled: First-Semester Abstract Algebra:
A Structural Approach, by: Jessica K. Sklar.
State if each of the below objects stated below is a well-defined set or not.
$\{z\in \...
- 4,532
1
vote
1
answer
114
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Comparability theorem for well ordered sets using transfinite recursion
Similar questions have already been asked here and here. But I am asking for verification of my proof. Halmos leaves out an "easy" transfinite induction argument, which I have struggled to ...
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