Tagged Questions

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Can someone give me some clue on how to show that rationals are well ordered? Thank you in advance.

I wanted to show that the set of rationals are well ordered. A small hint would be really appreciated.
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Are $1+ω$ and $ω+1$ isomorphic?

In set theory, $1+ω$ is defined as the ordinal number of ordinal sum of sets $\{a\}$ and ℕ. Also $ω+1$ is defined as the ordinal number of ordinal sum of sets $\Bbb N$ and $\{a\}$. You know what the ...
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Ingredients for Proving that a set is bounded below

I am having a hard time to really understand the definition of bounded below. The definition states : Let S be a nonempty subset of real numbers, we say S is bounded below if there is a c ∈ ℝ ...
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Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
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Proof of a variation of Hausdorff maximality principle

Let ($A$, $\le$) be a partially ordered set and let $B$ be a totally ordered subset of $A$. Prove that $A$ has a maximal totally ordered subset $C$ such that $B \subset C$. I'm trying to prove this ...
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order-isomorphism can send chain to other chain

exercise 7.20 in Goldrei: Let $a_1<\cdots<a_n,c_1<\cdots<c_n$ be rationals. Show that there is an order-isomorphism $f:\mathbb{Q} \to \mathbb{Q}$ such that $\forall i: f(a_i)=c_i$. ...
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Exercise in Well Orderings

Prove that if $\prec$ is a well-ordering on a set X and if $Y \subseteq X$, then $\prec_Y=\{(x,y) \mid (x \in Y) \wedge (y \in Y) \wedge (x \prec y)\}$ is a well ordering on Y. I am a little ...
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Set theory chain proof help!

Let $A_i$ be a partially ordered set and $C_i$ a chain of $A_i$. Order $A_1\times A_2$ lexicographically. Prove that $C_1\times C_2$ is a chain of $A_1\times A_2$. This is a set theory proof ...
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Prove that for every 2 elements in the set F of all functions from N to N, there's an element in F that's bigger than both

let there be $\ F$ the set of all functions from $\ N \rightarrow N$. K is a relation on F, for every f,g$\in$F , (f,g)$\in$K $\leftrightarrow$ for all $\ n\in N$, $\ f(n)\leq g(n)$ Prove that for ...
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How to show a quasi-order ~ on a set S with relation // induces a partially ordered relation?

Let ~ be the quasi-order relation. and let // be defined as the relation s~t and t~s(s,t elements of S). Show that ~ induces a partially ordered relation on the set of equivalence classes relation //, ...
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Building an antichain in a finite poset

Given some finite poset $P$ we would like to find an antichain $A$ which intersects each maximal chain. How to do that? Note that each chain $C$ and each antichain $A$ intersects at one element as ...