Let R1 and R2 be two relations defined as follows:
R1 = {(a, b) ∈ R2 : a2 + b2 ∈ Q}
and R2 = {(a, b) ∈ R2 : a2 + b2 ∉ Q}. Then,
1
R1 is transitive but R2 is not
2
R2 is transitive but R1 is not
3
neither R1 nor R2 is transitive
4
R1 and R2 are both transitive.