Let X be a set containing at least three distinct elements. Let S be the set of all non-empty subsets of X . Two relations R₁ and R₂ are defined on S as follows: A R₁ B A B A R₂ B A B for all A, B S . Then

Step-by-step solution

For relation R₁ : Reflexivity: Since A S is non-empty, A A = A . So, R₁ is reflexive. Symmetry: If A B , then B A . So, R₁ is symmetric. Transitivity: Let X = 1, 2, 3 . Let A = 1 , B = 1, 2 , and C = 2 . A B = 1 A R₁ B B C = 2 B R₁ C But A C = , so A is not related to C . Thus, R₁ is not transitive. Therefore, R₁ is not an equivalence relation. For relation R₂ : Reflexivity: A A is true for all A S . So, R₂ is reflexive. Symmetry: If A B , it does not imply B A . For example, 1 1, 2 but 1, 2 1 . So, R₂ is not symme