A relation on the set $\{1,2,3,4\}$ is defined as: $$\{(1,1),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,3),(3,4)\}$$
We're supposed to determine whether or not this relation is reflexive, symmetric, antisymmetric and/or transitive. I concluded that this relation is not reflexive, not symmetric, not antisymmetric and that it is transitive. Our Discrete structures mentor however claimed for this relation to also be antisymmetric, and this I don't quite understand.
According to the definition of antisymmetric two things must hold: if $R(a,b)$ and $R(b,a)$, then $a = b$,
or, equivalently,
if $R(a,b)$ with $a \neq b$, then $R(b,a)$ must not hold.
Because in our given set we have $(1,3)$ and $(3,1)$ I figured that this relation must in fact not be antisymmetric, as $R(a,b)$ and $R(b,a)$ both hold with $a \neq b$.
Is my definition of antisymmetry incorrect or did my mentor make a mistake?