You are given a discrete math problem. (a) Let the domain be ℤ. Consider S1: ∀x ∃y (x + y = 0 → xy ≥ 0) and S2: ∀x ∃y (x + y = 0 ∧ xy ≥ 0). Determine the truth value of each, and write the negation of each with no negation symbol in front of a quantifier or the whole formula. (b) Prove: for all integers a and b, if 3 divides a² + b², then 3 divides a and 3 divides b. (c) Let a₁ = 2, a₂ = 5, and aₙ = 3aₙ₋₁ − 2aₙ₋₂ for n ≥ 3. Prove by strong induction that aₙ = 3·2ⁿ⁻¹ − 1 for all n ≥ 1, and explain why two base cases are needed. (d) a₁ to a₅ are all prime. A student concludes that aₙ is prime for every n. Is this correct? Justify. Your task is to solve the given problem by providing a clear, step-by-step solution.

Prompt

You are given a discrete math problem. (a) Let the domain be ℤ. Consider S1: ∀x ∃y (x + y = 0 → xy ≥ 0) and S2: ∀x ∃y (x + y = 0 ∧ xy ≥ 0). Determine the truth value of each, and write the negation of each with no negation symbol in front of a quantifier or the whole formula. (b) Prove: for all integers a and b, if 3 divides a² + b², then 3 divides a and 3 divides b. (c) Let a₁ = 2, a₂ = 5, and aₙ = 3aₙ₋₁ − 2aₙ₋₂ for n ≥ 3. Prove by strong induction that aₙ = 3·2ⁿ⁻¹ − 1 for all n ≥ 1, and explain why two base cases are needed. (d) a₁ to a₅ are all prime. A student concludes that aₙ is prime for every n. Is this correct? Justify. Your task is to solve the given problem by providing a clear, step-by-step solution.