Symmetry in equivalence relationship is a→b, b→a.
(1) The only way I can interpret this question is by interchanging vertices:
E1={(1,2),(2,1),(3,2),(2,3)}
E2={(1,3),(3,1),(2,3),(3,2)} [vertices 2 and 3 in E1 change places]
E3={(2,3),(3,2),(1,3),(3,1)} [vertices 1 and 2 in E1 change places]
E4={(3,2),(2,3),(1,2),(2,1)} [vertices 1 and 3 in E1 change places]
So G1=(V,E1), G2=(V,E2), G3=(V,E3), G4=(V,E4).
(2)
E1={(1,2),(2,1),(2,3),(3,2),(3,4),(4,3)}
E2={(3,2),(2,3),(2,1),(1,2),(1,4),(4,1)} [vertices 1 and 3 in E1 change places]
E3={(1,4),(4,1),(4,3),(3,4),(3,4),(4,3)} [vertices 2 and 4 in E1 change places]
E4={(2,1),(1,2),(1,3),(3,1),(3,4),(4,3)} [vertices 1 and 2 in E1 change places]
E5={(4,2),(2,4),(2,3),(3,2),(3,1),(1,3)} [vertices 1 and 4 in E1 change places]
E6={(1,3),(3,1),(3,2),(2,3),(2,4),(4,2)} [vertices 2 and 3 in E1 change places]
E7={(1,2),(2,1),(2,4),(4,2),(4,3),(3,4)} [vertices 3 and 4 in E1 change places]
G1=(V,E1), G2=(V,E2), G3=(V,E3), G4=(V,E4), G5=(V,E5), G6=(V,E6), G7=(V,E7).
For transitivity, a→b, b→c⇒a→c. More to follow (I hope!)...