Prove that a relation is asymmetric if and only if it is anti-reflexive and anti-symmetric.
You have posted two problems. Who wrote these problems?
I ask that because both use non-standard definitions.
Does “anti-reflexive” mean "irreflexive?” If not then the proposition is false.
Your other question is meaningless unless $\displaystyle B \subseteq A$
I suspect that someone with a minimal knowledge of the area of mathematics wrote these.
Please review the question and correct any miss-typing.