you should know the difference between anti symmetric and A symmetric.
anti symmetric says - if (x,y) ∈ R then (y,x) ∉ R unless x=y [means it allow those pairs as well in which x and y are equal]
asymmetric says - if (x,y) ∈ R then (y,x) ∉ R [ it means he is more strict as he will not allow x=y pairs]
there are some relations which are antisymmetric but not asymmetric but there is no relation which is asymmetric but not antisymmetric.