All Questions
Tagged with order-theory proof-writing
67
questions
4
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0
answers
56
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Prove for a monotone, continuous, and rational preference relation $\succsim$ on $X=\mathbb{R}^L_+$, $y\geq x$ implies $y\succsim x$.
I need to prove the following result:
For a monotone, continuous, complete, and transitive preference relation $\succsim$ on $X=\mathbb{R}^L_+$, $y\geq x$ implies $y\succsim x$.
I tried it myself, ...
- 41
0
votes
0
answers
38
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Is the defintion given of a Hasse diagram correct?
I am currently reading Proofs: A Long Form Mathematics Textbook written by Jay Cummings, and am on the section about Partial Orders. When discussing how to visualize a POSET he defines a Hasse diagram ...
- 195
0
votes
0
answers
28
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If $R$ is a partial order and $a,b$ are incomparable, then the transitive closure of $R\cup\{(a,b)\}$ is antisymmetric
Claim: Let $T_0$ be the transitive closure of $R\cup\{(a,b)\}$. Then $T_0$ is antisymmetric.
Proof.
Note that since $R$ is transitive, $x T_0 y$ means that $x R a, b R y$ or $x R y$.
Suppose $x T_0 y\...
0
votes
0
answers
32
views
Is there a name for this ordering on integer vectors?
Let $\mathbf{k} \in \mathbb{[n]}^{u}$ be $u$-dimensional arrays, where $[n] = \{0,1,\dots,n\}$. Now let us assume that $\mathbf{k}$ are generated under $u$ nested for loops running from 0 to n. For ...
- 1,799
0
votes
1
answer
47
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I seek for a short and rigorous way to extend an embedding of a partial order into $\mathbb{Q}$
Let $(X,\leq ) $ be a partial order and $A$ be a proper subset of $X$ and $t\in X\setminus A$. Knowing that there exists an order-preserving map $\varphi :A\rightarrow \mathbb{Q}$ whose range is ...
- 375
0
votes
0
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31
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Property of magnitude proof
Is there a way to prove that:
If x < y and y = z, then x < z
I am trying to write a proof for my class but am currently limited to using Euclid's Common Notions and the Axioms for the Real ...
- 11
2
votes
1
answer
142
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Down-sets of a finite ordered set are down Max.
The following exercise comes from the book: Introduction to Lattices and order (second edition, B. A. Davey, H.A. Priestley).
Problem 1.14, pag. 27.
Let $P$ be a finite oredered set.
(1) Show that $Q=\...
- 1,278
1
vote
1
answer
87
views
Verifying Proof: If $L$ is a poset with a bottom element, and $\exists \sup(S)$ for every subset $S \subset L$, then $L$ is a complete lattice
I am currently working through Kaplansky for self study and was hoping to get some feedback on this proof. I would appreciate comments on clarity and legability as well.
Claim: If $L$ is a poset with ...
1
vote
1
answer
87
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Prove that $\mathcal{F}\cdot\mathcal{G}$ is the glb of the set $\{\mathcal{F},\mathcal{G}\}$
I want to refer to Exercise 26 part (c) of Section 4.6 of Velleman's 2nd Edition book.
The exercise is as follows:
Supose $A$ is a set. If $\mathcal{F}$ and $\mathcal{G}$ are partitions of $A$, then ...
- 338
0
votes
1
answer
50
views
How to prove this is a partial order? [closed]
A relation is defined by:
$x\leq y$ if and only if there exists $𝑘\in \mathbb{N}$ such that $y= x+5k$.
Prove that $\leq$ is a partial order.
I have no idea how to do this question. I've tried my best ...
- 11
1
vote
2
answers
419
views
Lowest upper bound of a family of sets
I refer to Exercise 23 of Section 4.4 of Velleman's 2nd edition book. It's also Theorem 4.4.11 in the Section. This post is not a duplicate of this (I include the posts that are mentioned there). I ...
- 338
4
votes
1
answer
267
views
Exercise 8.5.13 Terence Tao's Analysis 1
Background
I am trying to prove a lemma in Terence Tao's Analysis 1 which is used to prove the statement below.
Let $X$ be a partially ordered set with ordering relation $\leq$, and let $x_0$ be an ...
- 339
5
votes
1
answer
262
views
Prove that the least upper bound of $\mathcal F$ is $\bigcup\mathcal F$ and the greatest lower bound of $\mathcal F$ is $\bigcap\mathcal F$.
Not a duplicate of this or this.
This is exercise $4.4.23$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Prove theorem $4.4.11.$
Theorem $4.4.11.$ Suppose $A$ is a set, $\mathcal F\...
- 1,487
0
votes
2
answers
121
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Prove that if $x$ is the greatest lower bound of $U$, then $x$ is the least upper bound of $B$.
Not a duplicate of this or this.
This is exercise $4.4.21.c$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Suppose $R$ is a partial order on $A$ and $B\subseteq A$. Let $U$ be the ...
- 1,487
1
vote
1
answer
76
views
Suppose $L=\{((a,b),(a',b'))\in(A\times B)\times(A\times B)|aRa' \land(a=a'\rightarrow bSb'\}$. Show that $L$ is a partial order on $A\times B$.
Not a duplicate of this, this, or this.
This is exercise $4.4.9$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Suppose $R$ is a partial order on $A$ and $S$ is a partial order on $...
- 1,487