Question $6$Given:

$(1)$ Let $A$ be the set of real numbers, and $R$ be the equivalence relation such that

$xRy$ if and only if $x-y$ is an integer. Then, $[-1]=[1].$

$(2)$ If a relation on a set is not asymmetric, then it is also not antisymmetric.

Select one of the following choices:

a$.(1)$ is True and $(2)$ is False

b$.(1)$ and $(2)$ are False

c$.(1)$ is False and $(2)$ is True

d$.(1)$ and $(2)$ are True