For Exercise 4.3.15, you start by supposing a ∈ A ∪ B, and then have a ∈ A or a ∈ B, from which you could do division into cases. However, you also know a ∈ A or a ∉ A. Why? By negation (Game 1). Use that for division into cases. In the a ∉ A case you also know a ∈ B by elimination.