Question

is the relation antisymmetric?
A= {1,2,3,4,5}
R={(1,3),(1,1),(2,4),(3,2),(5,4),(4,2)}
the answer is no since (2,4) and (4,2) are in R. But I do not understand how.

Answer 1

A relation has the antisymmetric property when,
\[(a,b) \in R \Rightarrow (b,a) \notin R, \forall a,b \in A, a \neq b\]
There is another definition, but I think this one is easier to understand.
Another thing to remember is that, you can only have (a,b) and (b,a) in the relation if a=b.

Answer 2

Really? You think that way is easier to understand?

How about just:
A relation R on a set E is "anti-symmetric" if
\[\forall x,y \in E(aRb \wedge bRa \Rightarrow a=b)\]

Answer 3

That's the other definition that I was talking about, and it's equivalent to what I wrote: ', you can only have (a,b) and (b,a) in the relation if a=b.'
Both are correct of course, and that one is the more commonly used, but I finally understand the difference between symmetric, antisymmetric and asymmetric after I wrote the property the way I did in my previous message.
Of course that, if you are able to understand one, you should understand the other one.

Bottom line, everyone understand math in a different way, it's cool to have more than one way to refer to the same property :D

Answer 4

Yeah, I totally agree. In fact, it's probably more useful to think of the property your way when thinking about a relation as a set of 2-tuples (as the question was presented). Personally, I think the other way is more helpful when visualizing an ordering like so:



Anyway, great to hear your feedback!