1. Determine whether ∃ can be factored from an implication. In other words, is ∃x ∈ X,(p(x) → q(x)) ⇐⇒ (∃x ∈ X, p(x)) → (∃x ∈ X, q(x)) true? Explain your reasoning. Marks will only be given for your explanation. Hint: Think about giving meaning to each of X, p(x) and q(x).
2. Give a direct proof of the claim:
Let a, b and c be integers. If a|b and b|c, then a|c.