Strict Total Order for Real Numbers

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In summary, the relation of strict total order at the real numbers is antireflexive, trichotomous, and transitive.
  • #1
evinda
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Hello! (Wave)

A relation [tex]R[/tex] on a set [tex]A[/tex] is called strict total order if it is total order and satisfies the trichotomous identity, i.e. if [tex]R[/tex] is:- antireflexive, so [tex](\forall x \in A) \lnot(xRx)[/tex] or [tex](\forall x \in A) <x,x> \notin R[/tex]
- antsymmetric, so [tex](\forall x \in A)(\forall y \in A) (xRy \rightarrow \lnot yRx)[/tex] or [tex](\forall x \in A) (\forall y \in A) (<x,y> \in R \rightarrow <y,x> \notin R)[/tex]
- transitive, so [tex](\forall x \in A) (\forall y \in A) (\forall z \in A)(x Ry \wedge yRz \rightarrow xRz)[/tex]
- [tex](\forall x \in A)(\forall y \in A) \rightarrow xRy \lor yRx \lor x=y[/tex]According to my notes, the relation of total order at the real numbers is a relation of strict total order.[tex]\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}[/tex]But, in this case it will never be [tex]x=y[/tex], right? (Thinking)
 
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  • #2
evinda said:
- antireflexive, so [tex](\forall x \in A) \lnot(xRx)[/tex] or [tex](\forall x \in A) <x,x> \notin R[/tex]
This property is usually called irreflexivity.
evinda said:
- antsymmetric, so [tex](\forall x \in A)(\forall y \in A) (xRy \rightarrow \lnot yRx)[/tex] or [tex](\forall x \in A) (\forall y \in A) (<x,y> \in R \rightarrow <y,x> \notin R)[/tex]
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.[tex]\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}[/tex]But, in this case it will never be [tex]x=y[/tex], right?
What will never be [tex]x=y[/tex]? What precisely are you saying and why?
 
  • #3
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 [tex]x=y[/tex]? 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)
 
  • #4
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?
 
  • #5
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)
 
  • #6
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 $\forall x,y\in\Bbb R\;x<y$.
 
  • #7
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)
 
  • #8
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.
 
  • #9
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:

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

FAQ: Strict Total Order for Real Numbers

What is a strict total order for real numbers?

A strict total order for real numbers is a mathematical concept that defines a way to compare any two real numbers and determine which one is larger. It follows three main properties: trichotomy, transitivity, and anti-symmetry.

How is a strict total order different from a partial order?

A strict total order differs from a partial order in that it follows the property of trichotomy, which states that for any two elements, one must be greater than the other. In a partial order, this property does not apply, and it is possible for two elements to be incomparable.

What is the importance of strict total order for real numbers?

Strict total order for real numbers is important in mathematics as it provides a consistent way to compare and sort real numbers. It is also used in various applications such as computer algorithms, data structures, and database systems.

Can a strict total order be defined for other types of numbers?

Yes, a strict total order can be defined for other types of numbers, such as integers, rational numbers, and complex numbers. The properties of trichotomy, transitivity, and anti-symmetry still apply, but the way in which the numbers are compared may differ.

What is an example of a strict total order for real numbers?

An example of a strict total order for real numbers is the "less than" relation, denoted by the symbol "<". For any two real numbers, x and y, if x is less than y, then x is considered to be smaller than y in the strict total order. This relation follows all three properties of a strict total order.

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