(a) It is not symmetric since (a,c) ∈ R but (c,a) ∉ R
(b) It is symmetric but not antisymmetric.
Its by definition only.
a binary relation R on a set X is antisymmetric
if R(a,b) and R(b,a), then a = b,
or, equivalently,
if R(a,b) with a ≠ b, then R(b,a) must not hold.
So in option (b) (aRb) with a ≠ b and (bRa) is also holding,Thats why its not antisymmetric.
Ref : https://en.wikipedia.org/wiki/Antisymmetric_relation
(c) Its also not antisymmetric.
(d) It is satisfying both the properties.