Establishing if a binary relation expressed in set-theoretical notation is antisymmetric

A relation R is said to be antisymmetric iff (i.e., if and only if), for any pair a; b for which a is in the relation R with b and b is in the relation R with a, then a and b are equal:

\[R\quad is\quad antisymmetric\quad \Leftrightarrow\forall a,b \in A, (aRb\wedge bRa)\Rightarrow (a=b)\]

Practical tip: You can spot an antisymmetric relation in its set-theoretical notation by checking that there are no repetitions of couples of elements (even if in a reversed order) and that, at the same time, there is at least one element in the relation with itself.

Consider the following relation:

cRb, aRa, dRb, dRa, bRb, dRc

On the set A = {a,b,c,d}

The relation can also be represented in a set-theorical notation:

R = {(c,b), (a,a), (d,b), (d,a), (b,b), (d,c)}


Which of the following relations on the set A = {a, b, c, d} is actually an antisymmetric relation?