Yes, there is a relationship between symmetry and antisymmetry. Your conclusion is correct. If these three relations are all on the set x = {1, 2,3}, then:

R 1 satisfies reflexivity, symmetry and antisymmetry (R 1 also satisfies transitivity).

R2 satisfies symmetry (R2 also satisfies transitivity)

R3 satisfies antisymmetry (R 1 also satisfies reflexivity and transitivity).