1
vote
3answers
37 views

For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$?

For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$ ? Motivation for this question rose from an ...
0
votes
1answer
27 views

Proof by contradiction: $c=\max (A) \wedge d=\max (B) \to \max (A \cup B ) \in \{c,d\}$

Let be $A,B \subseteq \Bbb{R}$, with $A \neq \emptyset $ and $B \neq \emptyset$ and $\Bbb{R}$ totally ordered by $\leq$, and $c,d \in \Bbb{R}$, with $c=\max (A)$ and $ d=\max (B)$. I must proof by ...
1
vote
1answer
58 views

Are there any known uncountable transfinite increasing sequences of real numbers?

The real numbers are uncountable, so assuming the axiom of choice there is at least transfinite sequence of real numbers $r_0, r_1, r_2, ..., r_\omega, r_{\omega + 1}, ..., $ up to (and possibly ...
0
votes
1answer
50 views

Let $A$ denote a closed subset of $\mathbb{R}.$ Is it true that for all $B \subseteq A,$ if $B$ has a supremum, then $\mathrm{sup}(B) \in A$?

Let $A$ denote a closed subset of $\mathbb{R}.$ It seems obvious that for all $B \subseteq A$ we have that if $B$ has a supremum, then $\mathrm{sup}(B) \in A$. Perhaps this is trivial, but I cannot ...
2
votes
2answers
107 views

Dense and complete totally ordered sets

The totally ordered set $\mathbb{R}$ has the following properties. [Density]. Given any two $x,y \in \mathbb{R}$, we can find $\eta \in \mathbb{R}$ such that $x < \eta < y$. [Dedekind ...
0
votes
2answers
71 views

What is an order reversing bijection on $\Bbb R$?

What is an order reversing bijection on $\Bbb R$ and why must it be continuous? Thanks
1
vote
2answers
1k views

Prove that $\sup(A+B) = \sup(A) + \sup(B)$ and why does $\sup(A+B)$ exist?

We want to show that $\sup(A)+\sup(B)$ is the least upper bound of the set $A + B$. First, we need to show that $\sup(A) + \sup(B)$ is an upper bound for the set $A + B$. Indeed, if $z\in A + B$, then ...
2
votes
1answer
62 views

Equality and order in sets

Just started Baby Rudin and got struck in this. While defining order in sets, $<$ was introduced as a relation and for a set to be ordered the condition was: for all $x,y$ belonging to an ordered ...
2
votes
2answers
438 views

Infinum & Supremum: An Analysis on Relatedness

$\require{color}$ Question: I need some help in proving that $\color{green}{\text{if $k\geq 0$, then $\sup (kS) = k\sup(S)$ and $\inf (kS) = k\inf (S)$}}$, and also that $\color{red}{\text{if ...
2
votes
2answers
63 views

“intertwined” subset in linearly ordered set

I am trying to solve an exercise in utility representation theory which leads me to the following question. Take any (nonempty) linearly ordered set $(X,\succeq)$. Can we construct a set $Y\subset ...
0
votes
1answer
62 views

Definition of Dedekind cut as initial segment

the definition of Dedekind cut by initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B \neq \emptyset$, $B$ is Dedekind cut of $A$ under $\preceq$ ...
0
votes
1answer
64 views

Definition of initial segment

the definition of initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall b \in ...
2
votes
3answers
77 views

A basic question on the definition of order

In the first chapter of Rudin's analysis book "order" on a set is defined as follows : Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following two properties : (i) If ...
1
vote
2answers
97 views

inf and min in partial order

Assume $P$ is a partial order. Does it holds that $\inf (\inf(a,b), \inf(c,d)) = \inf(a,b,c,d) $? My guess is that it holds. But I don't know how to check it. Also what if instead of $\inf$ I use ...
3
votes
2answers
165 views

Motivation for a new definition of the derivative using the concept of average velocity

As everyone knows, for a function $ f: \mathbb{R} \to \mathbb{R} $ and a point $ a \in \mathbb{R} $, we say that the derivative of $ f $ at $ a $ equals $ L $ if and only if $$ \lim_{x \to a} ...
3
votes
1answer
301 views

Isomorphism from $\mathbb{R}$ to $(-1,1)$

There are many bijective functions that map $\mathbb{R}$ to $(-1,1)$, in particular: $$f\left(x\right)=\frac{e^{2x}-1}{e^{2x}+1}$$ (Of course there are others, such as ...
15
votes
6answers
931 views

Is the set of real numbers the largest possible totally ordered set?

Because I find any totally ordered set can be "lined up" in a straight line, I'm guessing that the set of all the real numbers is the biggest totally ordered set possible. In the sense that any other ...
3
votes
4answers
129 views

$\sup(S)$ does not belong to $S$?

Problem: can anyone come up with an ordering of $\mathbb{N}$ different than the standard one we know, where we can find a subset $S\subset \mathbb{N}$ in a way such that $\sup(S)$ exists in ...
5
votes
2answers
332 views

What does a well ordering of $\mathbb{R}$ look like? [duplicate]

Possible Duplicate: Is there a known well ordering of the reals? I am having a hard time wrapping my head around what a well-ordering of $\mathbb{R}$ looks like. I have seen the ...
2
votes
3answers
855 views

What is the difference between the supremum (least upper bound) and the minimal upper bound

I understand the difference between the supremum and the greatest element and between the supremum and the maximal elements. But I'm not sure about the difference between the supremum (least upper ...
2
votes
1answer
104 views

preorders induced by continuous functions to the reals

Any function to a (total) order induces a (total) preorder on its domain. What can be said about the total preorders induced by a continuous function from a "nice" topological space (for instance, a ...
4
votes
2answers
159 views

Does the set $\mathbb{Z}$ satisfy the completeness axiom?

The Completeness Axiom states: A set $\mathcal{A}$ satisfies the Completeness Axiom if for every of its non-trivial and bounded subsets, a supremum exists in $\mathcal{A}$. If this is the ...
3
votes
3answers
261 views

Is there a name encompassing both limit inferior and limit superior

Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?) The only thing I was able to find was that some ...
5
votes
1answer
730 views

Least upper bound property iff convergence of Cauchy sequences

From http://en.wikipedia.org/wiki/Non-Archimedean_ordered_field: The field of rational functions over $\mathbb{R}$ can be used to construct an ordered field which is complete (in the sense of ...