Let R₁ = (a,b) Z Z : ab 0 and R₂ = (a,b) Z Z : 3a + 4b is a multiple of 7 . Then on Z :

Step-by-step solution

For relation R₁ : Reflexivity: For any a Z , a a = a^2 0 . So, R₁ is reflexive. Symmetry: If ab 0 , then ba 0 . So, R₁ is symmetric. Transitivity: Let a = 1 , b = 0 , and c = -1 . We have ab = 1 0 = 0 0 and bc = 0 (-1) = 0 0 . However, ac = 1 (-1) = -1 Thus, (a,b) R₁ and (b,c) R₁ does not imply (a,c) R₁ . Hence, R₁ is not transitive and therefore not an equivalence relation. For relation R₂ : Reflexivity: For any a Z , 3a + 4a = 7a , which is a multiple of 7 . So, R₂ is reflexive. Symmetry: Let 3a + 4b = 7k for som