Antisymmetric Relation: How can I use the formal definition?

Question

So I can determine whether a certain relation is antisymmetric, by using a digraph. My understanding through a digraph is that if there is only 1 way streets and/or loops between edges, it's antisymmetric. But I don't know how that applies to the formal definition of a relation being antisymmetric, where $xRy$ and $yRx$ implies that $x=y$.

I'm going to take a fairly simple example.

$R = \{(a,b)(b,c)(a,c)\}.$

Now I do know that this relation is antisymmetric from making a digraph, but I'm not sure how the formal definition is applied here.

Thanks.

Answer 1

Here is a different, but equivalent definition for antisymmetry: For a relation to be antisymmetric, we need that for any element $(x,y)$ in the relation where $x \neq y$, the element $(y,x)$ must not be in the relation.

Now look at your example. Are there any $(x,y) \in R$ with $x \neq y$? Yes, all three, $(a,b), (b,c)$ and $(a,c)$. Do we have $(y,x) \in R$ for any of these? No, $(b,a) \notin R$, $(c,b) \notin R$ and $(c,a) \notin R$, so the relation is antisymmetric.

Answer 2

The "antisymmetry" statement looks like

$$\text{If } xRy \text{ and } yRx, \text{ then } x = y,$$

or, using the ordered pair notation,

$$\text{If } (x, y) \in R \text{ and } (y, x) \in R, \text{ then } x = y.$$

So, in your example, you just need to check if you have any ordered pairs $(x, y)$ and $(y, x)$ both in $R$: The only 'pair' of pairs allowed is $(x, y) = (x, x) = (y, x)$; that is, when $x = y$.

So, as long as there are no pairs $(x, y)$ and $(y, x)$ both in $R$, with distinct $x$ and $y$, then the relation is antisymmetric.