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We have a set of students.

$$ \{\text{Bob}, \text{John}, \text{Tom} \} $$

Scores of a exam were:

$$ \begin{align} \text{Bob} & : 5 \\ \text{John} & : 10 \\ \text{Tom} & : 10 \\ \end{align} $$

In the real life, we can sort by scores and say students are ordered.

$$\text{Bob} \le \text{John}$$ $$\text{Bob} \le \text{Tom}$$ $$\text{John} \le \text{Tom}$$ $$\text{Tom} \le \text{John}$$

But order relations are said to be antisymmetric.

$$ xRy \land yRx \Rightarrow x = y $$

In the example,

$$ \text{John} \le \text{Tom }\land \text{Tom} \le \text{John} \Rightarrow \text{John} = \text{Tom} $$

So, John and Tom must be equal, if the example is an ordered set. But it is only the scores that matches. If you compare the elements, they are different and John $\ne$ Tom.

Is it allowed to compare scores and say $\text{John} = \text{Tom}$? Can't we say that the example is an ordered set?

Adding some comment to make my question clear:

As nachosemu said, I am doing

$$ x R y \Leftrightarrow score(x) \leq score(y), \,\,\, x,y\in \{\text{Bob}, \text{John}, \text{Tom} \} $$

For this example, should I read antisymmetric relation as comparing elements:

$$ xRy \land yRx \Rightarrow x = y $$

or as $=$ in $\lt, =, \gt$:

$$ xRy \land yRx \Rightarrow score(x) = score(y) $$

Saigo Ueno
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3 Answers3

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In this case,

The (partial order) relation R is defined:

$$ x R y \Leftrightarrow score(x) \leq score(y), \,\,\, x,y\in \{Bob, John, Tom\} $$

After this, you are comparing this relation with another relation R':

$$ x R' y \Leftrightarrow x = y, \,\,\, x,y\in \{Bob, John, Tom\} $$

which is not the same one. Here, R' is an equivalence relation.

nachosemu
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  • Yes. That is what I am doing. So are you saying = should be, score(x) = score(y)? – Saigo Ueno Mar 16 '21 at 10:46
  • Yes, as you can see, 10 = 10. But R' leads us to say John $\neq$ Tom. It depends on how you define the relation. – nachosemu Mar 16 '21 at 10:47
  • That makes sense. But it makes antisymmetric relation vague because the meaning of = is not clearly defined by the rule itself. Do you have book or something that describes about it? It you be great if you have any. – Saigo Ueno Mar 16 '21 at 11:00
  • @SaigoUeno = really means they are the same thing. $10=10$, yes! They are the same number. $Tom=John$, no! They are not the same person. This is the same for all maths, there's nothing special about the antisymmetric relation or anything. – Milten Mar 16 '21 at 11:34
  • Sorry, but I'm getting confused. You answered yes, when I asked if = can be read as score(x) = score(y). Probably my question wasn't clear. Are you saying antisymmetric relation is $xRy \land yRx \Rightarrow x = y$ and not $xRy \land yRx \Rightarrow score(x) = score(y)$? – Saigo Ueno Mar 16 '21 at 11:44
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Starting with the bottom line: You can say either $\mathop{score}(\mathrm{John}) = \mathop{score}(\mathrm{Tom})$, or $\mathrm{John} \sim \mathrm{Tom}$ (not equal, cause they are different persons, but equivalent). Here $\sim$ is simply defined as meaning "having the same score".

You can also define an ordering directly on the students by $x\lesssim y\iff score(x)\le score(y)$ (this is basically what you're doing in the question). If you do this, what you get is in fact a total preorder on the set of students. A total preorder is the same as a total order, except it's not required to be anti-symmetric. But we can say $x\lesssim y\wedge y\lesssim x \iff x\sim y$. This is closely related to a total order (which is anti-symmetric): If you group the students into equivalence classes corresponding to points, you get a total order of two sets $$ \{\mathrm{Bob}\} < \{\mathrm{John, Tom}\} $$ As such, total preorders are total orders of equivalence classes. See also the top picture on the linked wiki-page.

Milten
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  • Thank you. If I can read antisymmetric relation as $xRy \land yRx \Rightarrow score(x) = score(y)$, then I think it will be total order. Are you saying the example is a antisymmetric relation or not? – Saigo Ueno Mar 16 '21 at 11:20
  • Antisymmetric means $xRy\wedge yRx \implies x=y$, you can't change that. So the order where you say $John \le Tom$ is not antisymmetric, because $John \ne Tom$. But you could say it's "almost antisymmetric" (not a real term) in the sense that $xRy\wedge yRx \implies x\sim y \overset{def}{\iff} score(x)=score(y)$. – Milten Mar 16 '21 at 11:29
  • And the fact that total preorders are "almost antisymmetric" is exactly why they correspond closely to total (antisymmetric) orders. – Milten Mar 16 '21 at 11:30
  • Thank you. I have clearly understood your answer. But the problem is that other people are saying different thing. If you know a book or sometnihg that describes about it, please let me know. Actually I have tried to find it, but couldn't find out. – Saigo Ueno Mar 16 '21 at 11:39
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The terminology "ordered set" does not really have a fixed meaning. This seems to be a perfectly reasonable confusion in the original question. It is especially confusing as the terminology "partially ordered set" does almost always mean the same thing. That is, a set together with a relation which is reflexive, antisymmetric and transitive.

There is no general agreement about what exactly an "order" is, and even more confusingly the word "total" can mean different things in combination with other words. For example, the difference between partial orders and total orders is not the same as the difference between partial functions and total functions even though all four kinds of thing are binary relations.

As other answers have already made clear, a given set can have lots of different binary relations. When modelling the world, more than one relation may capture something useful.

John Stell
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  • I could have used more precise word and sorry for that. But I don't mind whether it is partially ordered set or totally ordered set. I just want to know if the example is an antisymmetric relation. As written in https://mathworld.wolfram.com/OrderedSet.html, ordered Set is "An ambiguous term which is sometimes used to mean a partially ordered set and sometimes to mean a totally ordered set." And if you follow the links, you get to https://mathworld.wolfram.com/PartialOrder.html and https://mathworld.wolfram.com/TotallyOrderedSet.html . Both must be Antisymmetry. – Saigo Ueno Mar 16 '21 at 14:22
  • I did not mean at all that you chose the wrong word. What I was trying to get at is that the way language is used in mathematics is one of the challenges in learning new topics that is often ignored. The relation in the example is antisymmetric. The fact it looks odd to say John = Tom is not about mathematics, it's about the correspondence between the abstract model and the real world. John and Tom are different people with the same score. Ok to say John = Tom, but it only means same score, not same person. – John Stell Mar 16 '21 at 15:54
  • Thank you. But I'm getting an answer saying that the example is not antisymmetric. If you know a document describing about it, please let me know. – Saigo Ueno Mar 16 '21 at 16:04
  • It was just confusing what exactly the example is. Given the set ${J, T, B}$ and the relation ${(B,T), (B,J), (T,J), (J,T)}$, then that relation is not antisymmetric. – John Stell Mar 16 '21 at 16:11
  • Okay. Thank you. – Saigo Ueno Mar 16 '21 at 16:15