All Questions
Tagged with order-theory elementary-set-theory
713
questions
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Partial order on power set & set of partial orders
Consider a set $X$ and a partial order $\preceq$ on the power set $2^X$ of $X$. We assume that $\preceq$ extends the usual subset relation $\subseteq$, i.e. whenever $A\subseteq B\subseteq X$ then $A\...
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Why Is the Following Proof of a Finite Nonempty Totally Ordered Set Containing Its Maximum Wrong?
I wish to prove the result suggested in the title without induction on the cardinality of set. Here is my approach:
Let $S$ be a finite nonempty totally ordered set, i.e. $S=\lbrace x_{1},x_{2},\ldots,...
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2
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What are ordered pairs, and how does Kuratowski's definition make sense?
I have been watching the YouTube series 'Start Learning Mathematics' by The Bright Side of Mathematics.
I am currently on episode #3 of the set series and he's just introduced us to 'ordered pairs.'
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Binary subset rank and unrank [closed]
Let there be N=5 bits.
We want to rank and un-rank a specific subset of bits based on the following criteria -
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63
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Partial order on sets and application of Zorn's Lemma to construct well-ordered subset
I would appreciate help with the following question:
Let $(A,<)$ a linear ordered set.
a. Let $F\subseteq P(A)$. Prove that the following relation is a partial order in $F$: $X\lhd Y$ for $X,Y\in F$...
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Does $\langle\mathbb{Q},<\rangle\cong\langle\mathbb{Q}\times\{{1,0}\},<_{lex}\rangle$?
I recently encountered the following question on an exam, and I struggled to solve it. I hope to get some insight here.
Question:
Is the ordered set of rational numbers $\langle \mathbb{Q}, < \...
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24
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Szekeres example 1.5 errata?
In Peter Szekeres's text "A Course in Modern Mathematical Physics", example 1.5 (dealing with partial orders) says:
The power set $2^S$ of a set $S$ is partially ordered by the relation of ...
- 173
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75
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Law of Trichotomy for Well-Orderings
Often in beginning set-theory courses, and in particular in Jech's book Set Theory, it is proved from scratch that given any two well-orderings, they are isomorphic or one is isomorphic to an initial ...
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2
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113
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Is $\emptyset : \emptyset \to \emptyset$ an isomorphism from $(\emptyset, \leq)$ to $(\emptyset, \leq)$?
I was asked to determine whether the following statement is true:
If every function $F : P \to P$ is a homomorphism from $(P, \leq)$ to $(P, \leq)$, with $\leq$ an arbitrary order, then $|P| = 1$.
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How to prove "A finite saturated chain is maximal if and only if it contains both a minimal and a maximal element of the poset"
The wikipedia says:
Maximal chain. A chain in a poset to which no element can be added without losing the property of being totally ordered. This is stronger than being a saturated chain, as it also ...
- 416
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How do you prove that all countable, densely ordered sets without endpoints are isomorphic to the rationals?
I've looked online for a proof of this and have found several references to Canter's isomorphism theorem and the "back-and-forth method." However, I haven't been able to find any explicit ...
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One elegant proof of the transitivity part in "Show that lexicographic order is a partial ordering on the set of strings from a poset."
Recently, when I self-learnt Discrete Mathematics and Its Applications 8th by Kenneth Rosen, I did only the even-numbered exercises which the author offers the detailed description instead of the odd ...
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Topology as an order (not order topology)
Given a topological space $(X,T)$, the topology $T$ is also a partial order with the inclusion relation $(T,\subseteq)$.
Given a continuous function $f:A\to B$ between two spaces $(A,T_1)$ and $(B, ...
- 111
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2
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61
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comparing two tuple or set or list with greater than (>) or less than operator(<)
I have two set/list/tuple. A=(A1,A2,A3) and B=(B1,B2,B3). I know that each element of A is greater than B, meaning A1>B1, A2>B2 and A3>B3. How can I write this in correct mathematical ...
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Limit Countable Ordinal - is it a limit of a intuitive sequence of ordinals?
I am studying set theory, ordinal part.
Set theory is new to me.
I know that commutativity of addition and multiplication
can be false in infinite ordinal world.
$ \omega $ = limit of sequence $\, 1,2,...
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