MHB Strict Total Order for Real Numbers

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Hello! (Wave)

A relation R on a set A is called strict total order if it is total order and satisfies the trichotomous identity, i.e. if R is:- antireflexive, so (\forall x \in A) \lnot(xRx) or (\forall x \in A) <x,x> \notin R
- antsymmetric, so (\forall x \in A)(\forall y \in A) (xRy \rightarrow \lnot yRx) or (\forall x \in A) (\forall y \in A) (<x,y> \in R \rightarrow <y,x> \notin R)
- transitive, so (\forall x \in A) (\forall y \in A) (\forall z \in A)(x Ry \wedge yRz \rightarrow xRz)
- (\forall x \in A)(\forall y \in A) \rightarrow xRy \lor yRx \lor x=yAccording to my notes, the relation of total order at the real numbers is a relation of strict total order.\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}But, in this case it will never be x=y, right? (Thinking)
 
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evinda said:
- antireflexive, so (\forall x \in A) \lnot(xRx) or (\forall x \in A) <x,x> \notin R
This property is usually called irreflexivity.
evinda said:
- antsymmetric, so (\forall x \in A)(\forall y \in A) (xRy \rightarrow \lnot yRx) or (\forall x \in A) (\forall y \in A) (<x,y> \in R \rightarrow <y,x> \notin R)
Note that antisymmetry is implies by transitivity and irreflexivity.

evinda said:
According to my notes, the relation of total order at the real numbers is a relation of strict total order.\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}But, in this case it will never be x=y, right?
What will never be x=y? What precisely are you saying and why?
 
Evgeny.Makarov said:
This property is usually called irreflexivity.
Note that antisymmetry is implies by transitivity and irreflexivity.

So, if we want to show that a relation is strict total order, does it suffice to show that it is transitive, irreflexive and trichotomous? (Thinking)

Evgeny.Makarov said:
What will never be x=y? What precisely are you saying and why?

I meant that if $\langle x,y \rangle \in \langle_{\mathbb{R}}$ it will hold that $x<y$ but it will never hold $x=y$ or $y<x$, right?
So from the identity $(\forall x \in A) (\forall y \in A) \rightarrow xRy \lor yRx \lor x=y$ only this relation will hold: $xRy$. Or am I wrong? (Thinking)
 
evinda said:
So, if we want to show that a relation is strict total order, does it suffice to show that it is transitive, irreflexive and trichotomous?
Yes.

evinda said:
I meant that if $\langle x,y \rangle \in \langle_{\mathbb{R}}$ it will hold that $x<y$ but it will never hold $x=y$ or $y<x$, right?
Yes.

evinda said:
So from the identity $(\forall x \in A) (\forall y \in A) \rightarrow xRy \lor yRx \lor x=y$ only this relation will hold: $xRy$.
No, that's not correct. Are you claiming that no two numbers are equal?
 
Evgeny.Makarov said:
No, that's not correct. Are you claiming that no two numbers are equal?

I thought so, because of the relation we are looking at:

$$\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}$$

If $\langle x,y \rangle \in \langle_{\mathbb{R}}$ it means that $x<y$, right?

That's what I thought that we cannot find $\langle x,y \rangle \in \langle_{\mathbb{R}}$ such that $x=y$ or $y>x$, i.e. $yRx$.

Am I wrong? (Thinking)
 
evinda said:
If $\langle x,y \rangle \in \langle_{\mathbb{R}}$ it means that $x<y$, right?
Yes.

evinda said:
That's what I thought that we cannot find $\langle x,y \rangle \in \langle_{\mathbb{R}}$ such that $x=y$ or $y>x$, i.e. $yRx$.
Correct. But this does not imply that $\forall x,y\in\Bbb R\;x<y$.
 
Evgeny.Makarov said:
Yes.

Correct. But this does not imply that $\forall x,y\in\Bbb R\;x<y$.

I meant $\forall x,y \in R; x<y$, where $R=\langle_{\mathbb{R}}$.
This is right, or not? (Thinking)
 
evinda said:
I meant $\forall x,y \in R; x<y$, where $R=\langle_{\mathbb{R}}$.
This is right, or not?
This is a tautology (in the dictionary sense). But in post #3, you seemed to make a different claim.

evinda said:
So from the identity $(\forall x \in A) (\forall y \in A) \rightarrow xRy \lor yRx \lor x=y$ only this relation will hold: $xRy$.
Here you seem to be saying that for any $x$ and $y$, $x<y\lor y<x\lor x=y$ holds specifically because the first disjunct, namely, $x<y$ holds. This is wrong.
 
Evgeny.Makarov said:
This is a tautology (in the dictionary sense). But in post #3, you seemed to make a different claim.

Here you seem to be saying that for any $x$ and $y$, $x<y\lor y<x\lor x=y$ holds specifically because the first disjunct, namely, $x<y$ holds. This is wrong.

Oh yes, right.. (Nod) So, it will hold $xRy$ or $yRx$ but never $x=y$ where $R=\langle_{\mathbb{R}}$, right? (Thinking)
 
  • #10
Evgeny.Makarov said:
Are you claiming that no two numbers are equal?
...
 
  • #11
Evgeny.Makarov said:
...

How can it be that $\langle x,y \rangle \in \langle_{\mathbb{R}}$ and $x=y$ ? (Worried)
 
  • #12
evinda said:
So, it will hold $xRy$ or $yRx$ but never $x=y$ where $R=\langle_{\mathbb{R}}$, right?

evinda said:
How can it be that $\langle x,y \rangle \in \langle_{\mathbb{R}}$ and $x=y$ ?
You are shifting your claims from one post to the next, which is careless at best. Please write complete formal statements.

Edit: To clarify, by shifting claims I mean a shift from
evinda said:
So from the identity $(\forall x \in A) (\forall y \in A) \rightarrow xRy \lor yRx \lor x=y$ only this relation will hold: $xRy$. Or am I wrong?
in post #3 to
evinda said:
I meant $\forall x,y \in R; x<y$, where $R=\langle_{\mathbb{R}}$.
in post #7. So which is it:
\[
\forall x,y\in\Bbb R\;x<y\lor y<x\lor x=y\qquad(1)
\]
or
\[
\forall \langle x,y\rangle\in R\;x<y\qquad(2)
\]
where $R=\{\langle x,y\rangle\mid x<y\}$? And no, it does not follow from (1) that
\[
\forall x,y\in\Bbb R\;x<y.\qquad(3)
\]
And, yes, (2) is a completely different claim from (3).
 
Last edited:
  • #13
Evgeny.Makarov said:
You are shifting your claims from one post to the next, which is careless at best. Please write complete formal statements.

Aaa, I think that I got it now... (Wait)From the trichotomous identity, we get that $(\forall x \in A) (\forall y \in A) \rightarrow x \langle_{\mathbb{R}} y \lor y \langle_{\mathbb{R}} x \lor x=y$.

When we have two real numbers $a,b$, one of the above cases will necessarily hold.
It will be $a>b$ or $a=b$ or $a<b$.
I thought that the third case, $x=y$ is also related to the relation we have, but isn't.. (Shake) Right? (Thinking)
 
  • #14
evinda said:
From the trichotomous identity, we get that $(\forall x \in A) (\forall y \in A) \rightarrow x \langle_{\mathbb{R}} y \lor y \langle_{\mathbb{R}} x \lor x=y$.

When we have two real numbers $a,b$, one of the above cases will necessarily hold.
It will be $a>b$ or $a=b$ or $a<b$.
Yes.

evinda said:
I thought that the third case, $x=y$ is also related to the relation we have, but isn't.. Right?
I am not sure what you mean exactly here.
 
  • #15
Evgeny.Makarov said:
Yes.

(Happy)

Evgeny.Makarov said:
I am not sure what you mean exactly here.

From the trichotomous identity we have that $(\forall x \in A) (\forall y \in A) \rightarrow xRy \lor yRx \lor x=y$.
I meant that while $xRy$ and $yRx$ depend on the relation $R$ we consider, the third case $x=y$ does not depend on $R$.
 
  • #16
evinda said:
From the trichotomous identity we have that $(\forall x \in A) (\forall y \in A) \rightarrow xRy \lor yRx \lor x=y$.
I meant that while $xRy$ and $yRx$ depend on the relation $R$ we consider, the third case $x=y$ does not depend on $R$.
Well, yes, equality is the same in all laws of trichotomy (in the context of orders) while $R$ changes.
 
  • #17
Evgeny.Makarov said:
Well, yes, equality is the same in all laws of trichotomy (in the context of orders) while $R$ changes.

I see.. (Nod)
So, can we only find relations $R$ which are related to equality that satisfy this identity or are there also other trichotomous relations ? (Thinking)
 
  • #18
For all (finite, for simplicity) sets $A$ and $B$, $|A|<|B|\lor |B|<|A|\lor|A|=|B|$. Here $|A|=|B|$ is not the same as $A=B$. In general, of course, there are all kinds of situations where exactly one of three options necessarily holds.
 
  • #19
Evgeny.Makarov said:
For all (finite, for simplicity) sets $A$ and $B$, $|A|<|B|\lor |B|<|A|\lor|A|=|B|$. Here $|A|=|B|$ is not the same as $A=B$. In general, of course, there are all kinds of situations where exactly one of three options necessarily holds.

Could you give me an example of an other relation for which the trichotomy holds? (Thinking)
 
  • #20
I suggest you formulate this question precisely enough so that it would be suitable to start a new thread. Please also provide your attempt at solution, and it would be nice to include a rationale for such question.
 
  • #21
Evgeny.Makarov said:
I suggest you formulate this question precisely enough so that it would be suitable to start a new thread. Please also provide your attempt at solution, and it would be nice to include a rationale for such question.

I don't really have an attempt.. (Worried)
I was just thinking about whether it is possible that there is a relation that is not related to an equality or an inequality but satisfies trichotomy since one of its cases is $x=y$ or not... (Thinking)
 
  • #22
evinda said:
I was just thinking about whether it is possible that there is a relation that is not related to an equality or an inequality but satisfies trichotomy since one of its cases is $x=y$ or not...
At least state your problem precisely. I am not sure about the meaning of the following phrases:
  • "relation related to an equality",
  • "relation satisfies trichotomy",
  • "the cases of the relation".
 
  • #23
Evgeny.Makarov said:
At least state your problem precisely. I am not sure about the meaning of the following phrases:
  • "relation related to an equality",
  • "relation satisfies trichotomy",
  • "the cases of the relation".

We were looking at the relation:

$$\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}$$

We can easily see that the property would also hold if we would have the relation:

$$\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is greater that y}\}$$

Both of the above relations have to do with an inequality.

I was wondering if there is an other relation $R$, a one that is not related to $<$ or $>$ but satisfies the property:

$$(\forall x \in A) (\forall y \in A) (xRy \lor x=y \lor yRx)$$
 

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