All Questions
Tagged with order-theory solution-verification
180
questions
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48
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Problem with dense set
On ' Set theory with an introduction to real point sets'(Dasgupta, Abhijit ,2014) i found this exercise:
This is interesting because compare the topological (left,1) and order (right,2) definition of ...
- 387
4
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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
-2
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1
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57
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Prove that the order type of $\alpha\cdot\beta$ is the antilexicographic order in $\alpha\times\beta$. [closed]
This question is related to this one, but not a duplicate, since I am struggling with injectivity and monotonicity, rather than proving that $\{\alpha\cdot\eta + \xi:\eta<\beta\textrm{ and }\xi<\...
- 4,906
1
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1
answer
54
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A partially ordered set has all suprema iff it has all infima
Let $(P, \leq)$ be a partially ordered set. We will show that every nonempty set bounded above in $P$ has a supremum iff every nonempty set bounded below in $P$ has an infimum. Obviously, it suffices ...
- 1,649
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49
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Hasse Diagram multiple choice. Upper/lower bound and maximal/minimal.
Hi, this is one of the questions from my Discrete Mathematics exam that I got wrong. I believe I answered 2 since I did not see the "not" in the question.
Which of the following statements ...
- 9
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43
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Proof of non-isomorphic orders
Task: Let $A = \{(n, k) ∈ N × N : k \leq n\}$ and $B = \{(n, k) ∈ N × N : n \leq k\}$. Consider the restriction of the lexicographic order $N ×_{lex} N $to these sets: a pair $(n_1, k_1)$ is less than ...
2
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88
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Strict partial order and strict linear order
Task :
A binary relation on a set of 7 elements contains exactly 20 pairs.
Could it be :
a) a strict partial order relation?
b) a relation of strict linear order?
In strict linear order, any pair of ...
- 470
2
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0
answers
58
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Why is this description of the Dedekind–MacNeille completion never mentioned?
There are various ways to describe the Dedekind–MacNeille completion of a poset, the minimal complete lattice in which the poset can be embedded. I’ll first state the ones I’ve seen and then one I ...
- 239k
1
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36
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Does this bowtie shaped digraph define a semilattice in the sense of Hasse diagram?
I apologise for the bad formatting.
Motivation:
In my research, I have developed some ideas to do with semilattices. I regard them each as a set $L$ under an associative, commutative, idempotent ...
- 45.8k
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84
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Show that $\text{Part}(A)$ is a complete lattice
Let $A$ be a set and let $\text{Part}(A)$ denote the collection of all partitions of $A$. Define the relation $\leq $ on $\text{Part}(A)$ by $P_1\leq P_2$ if and only if for every $A_1 \in P_1$ there ...
- 4,827
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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
1
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1
answer
109
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Is this proof of the fact that any open set in an LOTS is a disjoint union of open intervals, correct?
This SE question invites proofs of the fact that any open subset of $\mathbb R$ is a countable union of open intervals. The highest-voted answer there is of Brian M. Scott where they give the ...
- 4,119
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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\...
2
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Proving a specific reccurrence relation. Proof Improvements.
Attempting to write proof for the given question. Are the proofs sufficient? Would appreciate any feedback and criticisms.
The relation $\star$ is defined on $\mathbb{R}-\{0\}$ by $x \star y$ iff $...
- 21
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For any $x$ in distributive lattice $L$, do we have $x = \sup \{a \in At(L) : a \leq x\}$?
Consider the following statement.
Let $L$ a finite distributive lattice. Then, for any $x \in L$, we
have $x = \sup \{a \in At(L) : a \leq x\}$}.
Let $X = \{a \in At(L) : a \leq x\}$. It is clear, ...
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