R = {(a,a),(a,b),(b,a),(b,b),(b,c),(c,b),(c,c)}
I know that to be anti-symmetric aRb and bRa, which this example has, but this example also means that a does not equal b (which anti-symmetry needs to be true). How, in this example, does a not equal b even though it has aRb and bRa.
This video tells me that aRb and bRa means a=b. But I also heard that antisymmetry needs there to be no edges (on a directed graph) going both ways (in other words, one-way) as shown here.
As you may have already worked out, I am very confused, and I just hope my question hasn't confused you. Any help would be much appreciated.
A relation is said to be symmetric when aRb if and only if bRa.
A relation is said to be anti-symmetric when aRb and bRa implies a=b.
Your relation is symmetric because it has (a,b) and (b,a) and (b,c) and (c,b).
Your relation is not anti-symmetric because it has (a,b) and (b,a) but a$\ne$b
(and (b,c) and (c,b) but b$\ne$c).
Another example of an anti-symmetric relation, besides the one given in the comments,
is "divides" for natural numbers, because, if a divides b, and b divides a, then a=b.