Consider the relation $R=\{(x,y)|gcd(x,y)=1\}$ on the set of positive integers.

Given:

$(1)R$ is reflexive and $R$ is antisymmetric.

$(2)R$ is symmetric and $R$ is not transitive.

Select one of the following choices:

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

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

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

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