1. Consider the relation j (divides) on the set Z. (a) Prove or disprove: j is reflexive. (b) Prove or disprove: j is symmetric. (c) Prove or disprove: j is transitive. 2. Assume R and S are two equivalence relations on a set A. (a) Prove or disprove: R [ S is reflexive. (b) Prove or disprove: R [ S is symmetric. (c) Prove or disprove: R [ S is transitive. 3. Consider the function θ : f0; 1g × N ! Z defined as θ(a; b) = a − 2ab + b (a) Prove or disprove: θ is injective. (b) Prove or disprove: θ is surjective. 1
