# Tagged Questions

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### Greatest lower and least upper bounds for a set of pairs

Had some trouble with this question in an exam recently, and wanted to make sure I reasoned correctly. The question was: $X$ is a set of pairs of real numbers $(x,y)$, with absolute values less than ...
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### Have you seen this property of tolerance relations before?

Let $A$ be a set equipped with a binary reflexive and symmetric relation $\uparrow$ (such relations are often called tolerances, see also "Are there real-life relations which are symmetric and ...
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### Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
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### what kind of relationship is “is prefix of”?

Consider the "is prefix of" relationship on a set that corresponds to the words of some alphabet. E.g. "ab" is prefix of "abc". This relationship is: antisymmetric transitive reflexive ("ab" is a ...
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### Is there a relation that is irreflexive, anti-symmetric and not transitive?

from the set $\{a, b, c, d\}$? Of the one's I have tried, it at best is two of the three, but never all.
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### Question about sets of well-orders and isomorphisms between well-orders

I was thinking about this problem in trying to better get a feel for and understand well-orders (I'm trying to learn some set theory) - right now, I'm not really hoping for anybody to supply me with a ...
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### Proof that there is an order relation

For an arbitrary set M there is a relation $R \subseteq 2^M \times 2^M$ about $$A \mathrel R B \Leftrightarrow A \cup \{x\} = B$$ The join is a disjoint join. There are not more details what is $x$. ...
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### Total Order Relation

Given $X = \{ a , b , c \}$ and $R = \{ (a,a) , (b,b) , (c,c) , (a,b) , (a,c)\}$ Is $R$ a total order on $X$? I know that total order requires the relation to be comparable on all elements, ...
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### I want to show that $\mathrel{R_1}$ and $\mathrel{R_2}$ are a partial order on $\mathbb N$, how would I do this?

Let $\mathrel{R_1}$ and $\mathrel{R_2}$ be relations on $\mathbb N$ defined by $x\mathrel{R_1}y$ if and only if $y = a + x$ for some $a \in\mathbb N_0$. $x\mathrel{R_2}y$ if and only if ...
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### How to tell if relation on set is a partial order when relation is defined as a set of ordered pairs?

Determine whether $R$ is a partial order on set the $S$ and justify your answer. $S=\{1,2,3\}$ and $R=\{(1,1),(2,3),(1,3)\}$. So the job is to consider if $R$ is reflexive, antisymmetric and ...
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### Finding order properties in the relation $aSb \iff \exists k \in \mathbb{N} : b = ak$

An order relations exercise I just did. I think it's fine, but the second proof felt a bit too wordy or discursive, instead of going straight to the point with brief and accurate statements. How could ...
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### Finding the first, last, minimal and maximal elements in these relations.

I'm now learning about relations of order. This is what I gather: First element: Precedes all elements Minimal elements: Have no predecessors. The first element is always a minimal. Last element: ...
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### Is = (equality) a partial order relation?

We know that a partial order relation is a relation which is reflexive , antisymmetric and transitive. Example: (x,y) belongs to R iff x=y. For A={1,2,3}, we get R= {(1,1), (2,2), (3,3)}. Now R is ...
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### Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
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### Proving well ordering is total relation

Assuming R is well-ordered relation on a set A. Wikipedia claims that every well ordered relation is also a total one.Even tough in first sight it seems trivial, how can I prove it?
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### Show that a relation is a partial ordering

Show that the relation $R$ on $\Bbb N$ given by $aRb \text{ iff } b = a2^k$ for some integer $k\ge 0$ is a partial ordering.
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### Union of two partial orderings

Suppose S and R are partial orderings. Does is necessarily mean that $R \cup S$ (union) is a partial ordering? If not what conditions would have to be met for it to be a partial ordering?
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### Composition $R \circ R$ of a partial ordering $R$ with itself is again a partial ordering

If $R$ is a partial ordering then $R\circ R$ is a partial ordering. I cannot seem to prove this can anyone help ?
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### Name of the order with irreflexivity, antisymmetry and transitivity?

I have an order otherwise poset aka partial order but it is irreflexive so relationships such as 1R1 and 2R2 are impossible. What is the name of this order?
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### A generalization of Galois connections

Let $f(x)\ast y \Leftrightarrow x\ast g(y)$ for some binary relation $\ast$ and functions $f$ and $g$. This is a generalization of Galois connections. Are things like this studied before? Note that ...
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### “Lexicographic order” without priority, but with ties, how to define / what's the name?

I'm searching for an ordering principle which combines two (or more) preorders, akin to the lexicographic order, but not quite: I'm looking for an ordering principle that does not priorities, but ...
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### Is lexicographic order complete?

I was wondering, if the lexicographic order in $\mathbb{R}^2$ is complete or not? I guess it isn't but I dont find any counterexample. Complete means: For every partition of $M$ into two disjoint ...
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### How is the Power set without the original set a partial order?

This is a past exam question. Let S = {a,b,c} Define $\mathcal{P}(S)$ as the power set of S. Consider $\mathcal{P}(S)\setminus S$ Explain how this structure illustrates the concepts of partial ...
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### Can a Partial Order be symmetric in addition to its properties?

For example,a relation,R defined on integers. $R = \{(a,b)\mid a^3=b^3\}$ Could this be partial order?
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### Proof for a creating a partition of a countable set using chains in partial orders.

Definition: A partition of a set $A$ is a set of nonempty subsets of $A$ called the blocks of the partition, such that  every element of $A$ is in some block, and  if $B$ and $B'$ are different ...
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### The proper subset relation and strict partial order

The proper subset relation, $\subset$, defines a strict partial order on the subsets of $[1,6]$, that is $pow[1,6]$. (a) What is the size of a maximal chain in this partial order? Describe one. (b) ...
I'm having trouble figuring out how I can solve this... I've never been good with formal proofs. $$(\mathbb{R},\preceq), a\preceq b\iff a^{2}\leq b^{2}$$ I can easily see that it's Reflexive: ...