What is Anti-Symmetric Relation?

[iwantabelieve]

Hi guys.

Here I have a question about how to understand the definition of the anti-symmetric relation.

First, we have the following:
If a R b and b R a, then a = b, which is the definition of anti-symmetry.

What I want to know is whether those ≤, ≤, and = are in their literal sense, or something else.

For example, suppose S is a set of alphabetical words, while a and b are two of its members. If we want to express a is no longer than b, we can write: a ≤ b, and b no longer than a: b ≤ a. Here comes the problem, if a=b means a has the same length as b, then anti-symmetry holds for it; however, if a=b means a and b have to be the same word, then anti-symmetry does not hold for this. If a and b are numbers, I think I know the answer well; however, in this specific problem, they are words. So I do not know if I should take ‘=’ in its literal sense, or I should take it as meaning ‘equal in length’.

Any idea is welcome. Thank you in advance!

 

[HallsofIvy]

A relation is "symmetric" if, any time you have aRb you also have bRa.

A relation is "anti-symmetric" if, any time you have aRb you cannot has bRa.

Of course, most relations are neither symmetric nor anti-symmetric- you have aRb and bRa for some a and b but not all.

As for your example, I think you have it exactly backwards. If aRb means "a and b have the same length", then R is symmetric, not "anti-symmetric". For example the strings a= "xyz" and "uvw" have the same length so aRb is true- but then of course, bRa is also true. If aRb means "a and b are the same word" then R is also symmetric, not anti-symmetric.

If aRb means "a has more letters than b" ($a> b$, then if aRb, it is impossible that bRa. That is an anti-symmetric relation. If aRb means "a has at least as many letters as b" ($a\ge b$), then with a= "xyz" and b= "uvw" we have both aRb and bRa so R is NOT anti-symmetric. But if a= "pqrs" and b= "xyz" then we have aRb but not bRa so R is not symmetric either.

 

[scrambled]

HallsofIvy - you are confusing antisymmetry with asymmetry. As iwantabelieve said:

If a R b and b R a, then a = b, ... is the definition of anti-symmetry.

So the question is, what does the "=" mean? Is it identity or equality?

In short, it's equality. So "...is no longer than..." is indeed antisymmetric in the domain of words (or strings for that matter).

I see your problem. Every definition out there just uses "=" without specifying and, generally, examples are from the naturals or reals where there are no equal but non-identical pairs. Hmmm... are 0.999... and 1 identical as well as being equal? I guess not, in which case ≤ being antisymmetric in ℝ (a common example online) demonstrates that the "=" is not identity.

I know this is way out of date but I was wondering about this myself recently and it's a good question with no quick answer online.